SO the question is about "helping" a group makeup that is lacking something "essential" and what folks might suggest.  I will start with a specific example (a group I am playing in) and offer a variant, so maybe a broader range of ideas can spring forth.

Our group consists of a Dwarf Barbarian, Teifling Druid, Human Fighter and Genasi Monk (me)  Obviously we are rather seriously UNDER magic-ed.  Our Druid is Land, if that helps with suggestions, I am Open Hand, Fighter went Champion and the Barbarian is Berserker.  All but the Fighter are relatively knowledgeable players, although we proved we are careless in not balancing the party well to begin.  I blame me primarily, as I usually choose last and overestimated the Druid's spell list.  We've reached 5 and are discussing multi-classing a couple levels to make up for our missing bits.  Druid is looking at Wizard, to give her more options at range.  I am considering Cleric, to ensure we have some heals.  Fighter is kind of new-ish, so we're not counting on him and our Barbarian Max statted, so he can go Rogue or Fighter, neither of which helps us much, really.

SO we got a wand last session of Eldrich Blast, which is a nudge to allow our Druid to keep back some, I think.  It gives her a "free" damage bolt, which is nice.  I think the Wizard multi is still a good idea, to again, expand her damage potential and add some extra utility (Identify, Find Item, etc) so it makes sense.  I, on the other hand, feel I can best help the group by dealing damage, interfering with stronger foes (stun/knock prone/kill reaction) and basically being a spider monkey when we need to get stuff.  I was looking for healing potions in the equipment manager and noticed the Staff of Healing (I am way too N00B to linky) which would address my only REAL reason for wanting to multi.  Yes, a Cleric gets some truly helpful little buffs early, but I would need to dive a few levels in to get enough spell slots to be of much use.  This delays getting more utility stuff from the main class, like running up walls, evading AOE damage and so forth.  I am really torn as to what would be the most feasible solution and I am interested in others opinions on how best to tackle it.

Second idea is one a friend asked me about.  He is running a session and the group brought a Ranger, a Wizard, a Druid and a Rogue.  Ranger's apparently using a bow, so NOBODY is geared to go stand in front of the meanies and get punched in the face.  He's been trying to gear encounters where ranged exchanges are more relevant, but this is leaving the Rogue a little left out. 

For what it's worth, I feel HIS group is in much worse shape than my own, because on the couple "boss fights" we've had, the 3 melee have been able to unload points/slots/skills and drop the foe before we get too smashed up.(I did die once, but the action killed the baddie too, so I got Revivified with a scroll....which launches into a looong story about my Monk finding religion)  Our Barb is getting enough beef to be able to take the hits and our Fighter just got a new shield to reach 19AC, so while we are magic starved, anything that is melee is getting it's ass kicked rather handily.

Sorry....I babbled.....again.

This is a tough choice, personally I think odd groups could be really fun to play but if you're having a ton of issues then adjusting your composition makes sense. If you want more healing but don't like the idea of multi-classing out of monk then you could switch your subclass to the Way of Mercy. This would give you some additionally healing and you'd be on track to get your evasion at a normal level - using your example of your monk finding god could easily be used for the subclass switch as well as multi-class into cleric. If you still need more healing you could still dip into cleric later, or if your DM is super nice they could just give you a homebrew Staff of Healing that also counts as a monk weapon.

What might work better for the second example, although you could use it for your group as well,  is using a sidekick. Maybe have one of the more experienced players whos playing a bit of a simpler character run the sidekick in combat. The group could jointly run them for RP purposes if no one wants to run a second, albeit streamlined, character. 

5e is actually pretty forgiving of team comp. You can have a successful party of almost any makeup, so I wouldn't say it's absolutely necessary for anyone to multiclass. If you find yourselves low on healing, just stock up on a few potions that can carry you over between short rests.

So, as I understand it, you are asking what needs to be done about the following party combos:

Party 1: Barbarian, Fighter, Monk, Druid

My suggestion: Nothing needs to be done. There is no reason you need an "arcane" caster. The druid will be able to cast plenty of cool spells as they level. And you have THREE frontliners to hold off enemies while the druid does their thing. I see no issues with such a party. Will it play like a standard "Holy Trinity" part of Fighter, Wizard, Cleric, Rogue? No. But the only thing "Holy" about RPG "Holy Trinity" class make-ups is the reverence with which narrow-minded people who only use boring, unoriginal, inside-the-box thinking hold them. You could have a party of 4 monks and still have a blast, have fun, and defeat most encounters.

Party 2: Ranger, Wizard, Druid, Rogue

My suggestion: Nothing needs to be done. You do not need a "tank." Is it easier if you have one? Yes -- assuming you decide to again, think inside the box and use "standard" game tactics instead of some creativity and smart thinking. My current group of PCs is Ranger, Sorcerer, Rogue/Fighter, Cleric - nobody has very good armor (the cleric is an Aarakocra and wants to stay in light armor for flight, and the Rogue/Fighter is more Rogue than Fighter and has a Chain shirt, and the Ranger is a beastmaster). There isn't really a tank. They're fine Heck, in one encounter the Sorcerer basically served as the tank... He tried to Hold Person on the boss, the boss saw him as the threat and kept trying to hit him, and he kept spending his action disengaging and getting away. Now yes, he didn't do a lot of damage, but he soaked all the boss attacks until the party could surround the boss and take him down. Nobody's HP dropped to zero, against a pretty tough boss (and their first Legendary enemy).

I submit to you that the idea that these parties, which are made up of 4 different character classes, are somehow "sub optimal" or "not viable" comes from the tendency of many gamers to get stuck in a rut, to stick with "what works in the most obvious way". There's no need to do that. Any class combo can be made to work if players learn their class well, play intelligently, and think before they act.

Heh - I remember one game (2ed days) where the ranger expected me as the wizard to tank. Other player was a rogue IIRC. We were all level 1. Oh, and he was using a bow against skeletons. 

But I digress - I am a fan of "broken" character parties, but you do need the players and the DM to be able to think outside the box and let them exist. In the first example you have the option of herbalism and make your own potions. You could also (later ... much later) lean into the no magic and have semi-permenant silence effects or anti magic effects on yourselves. But mostly it takes a shift in mentality away from "go there kill everything". I fall into the "trap" too, but you don't need to. Often the goal is something like "retrieve this magic item". And so you fight your way through a dungeon, arrive at the last room in a wash of blood and kill the boss and take the item. But you could just join the cult and filch the item while you are "polishing it" one day. You could stage a rival cult or do some crimes which implicate this group - and so turn the populace or common folk against them. Yes it because different from your usual dungeon crawl but it is no less (and I would say often more) fun than a usual hack and slash. 

The beauty of D&D is that it isn't a video game and the path doesn't have to be hard coded. Unfortunately it is hard to think on your feet and allow something reasonable to work if it is a surprise to you. 
I hate the healer/tank/2-3dps need that MMOs have brought to the gaming mentality. It is a way, but it doesn't have to be the only way.

It's not up to you to balance the party. It's up to the GM to create adventures that will be suitable to the group.

Yeah, I can only tell you as a DM that you can write for your players.  If he CHOOSES to shred you up with missile weapons instead of telling a boss story about this young druid and his burly protectors, you can decide whether to adjust strategies or tactics while rolling up your replacements.

Hmm, patience seems best, lol.  My monk can, as said, go Cleric or Mercy Monkey and follow a sensible reasoning and our DM is certainly on board, shifting things subtly to try to ensure our survival.  Maybe it's just the memories of my Bard coming through, where there were no gaps in what we could do as a group.  Seemed every base was covered, and my Bard had a rather odd array of spells to ensure it.

We are trying to stock up on potions to limit abuse of the "short rest" all the time.  We've also recently gotten a few trinkets to boost our AC which should be a big help.  Last fight, we felt pretty confident, but the foe rolled a 20 and a 23 on his attack rolls, pounding our fighter pretty hard.  While I realize there's no accounting for big rolls by the baddies, having a decent sized pop heal would be nice.  Druid is starting to use a few Healing Word if things get sketchy to keep us upright.  Expensive use of a spell slot though.  MMM will adopt the patience path for now.  Moving from Open Hand doesn't work, as on bigger enemies I have been stunning or knocking them prone.  I can also make them lose their reaction for the round, allowing me to simply move 40 feet out without fear of opportunity attack, since our bigger foes can swat me on a solid double attack sometimes.  Without doubt, playing a non-ideal group is a lot of fun and challenge.  We recognize our strengths when a fight breaks out and tend to blitz target to target since melee damage output is our greatest strength.  The fight our Fighter got bashed up, the DM expected to be a 5+ round scrap.  By using an item (and causing myself 10 damage) I smashed it hard to open, then the Fighter ripped off, Barbarian behind him and our Druid....cast Haste on the Monk.  Round 2 I knocked him prone on first attack round with a crit and got another crit on the second round which....ended our enemy's days.

Dammit, babbling again

It seems like you are doing fine. The chaos you describe is what happens in normal D&D. 

An idea, if you don't hate the idea of homebrew? Try substituting jury-rigged technology for some spell effects. 5e has more support than at first glance for replicating a lot of spell effects with gear if the players are creative and those systems get expanded a lot by lots of the homebrewers at DMsguild ("City and Wild" is I think a popular one that has a lot of expansions to crafting, potioning, and poisoning, as well as straight-up purchasing spell scrolls which, by the by, can be used by anyone. On the other hand, if you buy too much into its systems, you may end up playing a 4x strategy instead of an RPG, so, caution advised I suppose?)

It can be a good bit of extra work for a DM, but coming up with "creative" concepts to replicate spell effects with "mundane" "magic items" is something I have fun with in my spare time, and is pretty easy to balance the whole "no spell slots" thing simply by making players track ammo/perform daily maintenance for the gear, etc.

Or, if the players don't think it's a problem, maybe it isn't. I came to this site just to receive that same advice once, haha.

I literally have a multiclass rogue/barbarian/wizard right now. Anything is possible. Also, multiclass into wizard if only because of find familiar.

All good DMs play to their audience. They don’t force PCs into bad encounters or encounters that they feel frustrated with. So I’d say talk to the DM and make sure he’s fine with your group comp and not trying to “game” or so he can “win” encounters. 

Also, as a player don’t meta game your character to “cover holes” either. That takes me right out of my character narrative and makes it less fun to play the character.

I once DM’d for a group that was 2 rogues and 2 barbarians. It was fantastic fun because I left the magic level low and gave them enemies that could beat them on their own terms if they didn’t work together. 

I wouldn’t really worry about much honestly. One time I had a party that was half rangers and no casters and we turned out ok. Until the Paladin died. So I think you should be good.