I've just started out as a new DM with a group of friends all of which are new to D&d, I let them have free reign with their characters but said that it would be a good idea for atleast one person to have some healing abilities (which of course no one listened too...) we started the lost mine of phandelver and immediately ran into the issue that they were being beaten bloody and continually having to take short rests. Would it be wise to introduce a DMPC as a dedicated healer into the party if they just stuck to the background and just followed?? 

Why not? If it's not a problem for you and everyone gets to have fun.

In my group's campaign, no-one wanted to play heals or tank so, I get to play both Cleric and fighter. Everyone else plays their single damage class character, all other players and the DM are fine with this. We hope to get a Barbarian player, then I will focus on Cleric.

It seems really common to me for the DM to add helpful npcs as a resource to fill in party needs. If everyone likes gritty combat and surviving by medicine kits, potions and hit dice recovery, then do that.

Let the players figure out what to do. When I played in a group like that I was always buying healing potions so we had healing. We made do. 

Just let them find a chest of healing potions, or a note that suggests that certain mosses on the walls of the cavern can be consumed for a rush of healing. (Add a "use-by date" if you're worried on them stocking up too much on it.

You'd be forcing the DM-PC in anyway, so these suggestions aren't too immersion-breaking.

Rather than forcing a DMPC healer on them, ask if they want to hire a healer like an Acolyte, bard, or priest.

You can use optional rules such as the Healing Surge, so you don't have to change group composition nor add NPCs.

Or you can add a Sidekick (Unearthed Arcana, or Essentials Kit). UA is playtest or unofficial material, EK is published official material.

If you want them to have more healing, just have them find a few more healing potions than you'd typically give. Or just let them take those short rests. It's fine, as long as they're having fun.

If the story is just starting & they are starting to feel the absence of a healer, maybe allow someone to re-class to paladin and/or reveal Aasimar heritage?

Alternatively; it isn't much of a hop from most professions to a cleric domain, if someone wants to reveal their rogue was actually on a mission for a trickster for instance - though ability scores would need some reworking.

As they are new to D&D and just starting out, I would lean that direction over multi-classing.

Each DM has to learn what's best for their own gameplay ass the group. If you feel that your party needs a healer ask why? That's why we have healing potions. So maybe they can spend some of that gold they earn to buy potions.

Until an actual healer joins the party they should have to depend on potions. Alternatively to make things easier on you each town could have a healer for hire how you see fit.

In my group the NPCs which I get to play all have good parts in the campaign. For example a pirate who they never checked or asked about was a high level cleric they left him behind. Another was a high level alchemist who they got killed and was the only one that could have ended a curse.

My players have always asked the high level NPCs I bring to them why they don't help as much as they could. My answer is always.. well the npc tells them that this is their story, their adventure. They'll be there when they really need it but this is not the time.

I wouldn't want a DM to play a character. For me it takes things away from what was to be learned. The hard way if necessary. That's just me though.

When I started as a DM I had a Life Cleric as a DMPC. Any rewards and loot were always to the party first and suited for them, my character would help prompt conversation but never lead the discussions or vote on courses of actions and provided healing in combat - mostly he was there in case I made an encounter too difficult and so the group did not need to focus on healing if they did not want to. He was a useful tool being rather inactive in a lot of things and only used sparingly as I needed. If a player could not make it, I could then have him take a more active role.

DMPCs can be a mixed bag. They can be very useful tools but some DMs may play them too much like a player and end up making them the star or getting the best loot or face challenges that their character just-so-happened to be the best at (been in games where the DM did this, it was not fun at all). However, the DMPC can be used well - I'm quite good at differentiating what I know and what a character would know and ensuring every player gets their shot. Even as a player, where an enemy used Mislead, I know the enemy was not really there but I also know my character would not have any reason to suspect anything different and willingly wasted my highest level spells knowing they would fail because it's what the character would do in that situation. As a DM, if you're going to have a DMPC be sure you know how to not metagame yourself and to ensure your DMPC will always take backseat to any actual PC - your DMPC will not have some big story arc to take the spotlight, they should not be geared to handle the situations the party faces but also you should not avoid the sensible traps and stuff just because your DMPC has a way to mitigate it (just because your cleric can easily cure poison doesn't mean you should avoid poison traps if a poison traps makes the most sense). My DMPC was only to serve two purposes: emergency healing in combat, RP conversations to ensure everyone feels like they have a chance to RP. This is mostly because I was new and he was my safety net and also because I had healing potions be harder to come by (it doesn't make sense to me for healing potions to be so easy to come by: if they could be so easily obtained, why wouldn't the enemies also have them?).

DMPCs are very useful tool but should be used with caution.