I really do like the Path of the Beast conceptually (growing natural weapons and moving on the ceiling cinematically!), but the Form of the Beast just doesn't strike me as viable as-is as the character levels and gains access to magical weapons...

• Bite: peaks at level 1 when 3-5 health represents significant healing, then all downhill from there. Technically it remains use-able throughout - as you can (I believe) make attacks within your Attack action using different 'weapons' to get the one-bite-per-turn in - but as soon as you get your first +1 weapon or weapon-boosting feat it isn't a competitive option to do so.
• Claws: this is definitely the most viable option, and I confess the full maths is beyond me (one versatile weapon attack + two claw attacks w/ so many variables) but my gut instinct is that level 9 is when it fizzles out; Brutal Critical comes online (1d6 is quite puny), there have been 2-3 opportunities for a feat & magical weaponry is a near certainty - even with Rage Damage, I think the appeal of claws diminishes from this point.
• Tail: so long as you are prepared to use a giant tail as your weapon, this is great up until your first +1 martial weapon (or weapon feat) & will then be little used after. If you embrace the scorpion-person motif it does have a niche long-term - delivering Infectious Fury till you run out of uses & poking immobilised foes outside 5ft weapon range - but always as a complement to your primary weapon attacks.

Infectious Fury is undoubtedly a neat feature, but it's not a game-changer in terms of natural weapons vs magical great-weapons as it will undoubtedly fail to land many times at high levels - combination of no attack bonus & wisdom saving throw.

My objectives...

1. Allow the barbarian to go weaponless - rage isn't constant, particularly at low levels, and I'd like to avoid a repeat of the Lycan (carrying a greatsword but making unarmed strikes).
2. Provide a bonus action outlet - the only way I see for concepts like Bite to be viable is to not have them compete with regular weapon attacks, so bonus actions seem apt.
3. Find a way for low damage die attacks to remain relevant in a class that has huge criticals as a central mechanic.

My suggestion...

Form of the Beast 3rd-level Path of the Beast feature

When you enter combat, you can transform, revealing the bestial power within you. Until combat ends, you manifest the following natural melee weapons:

• Bite. Your mouth transforms into a bestial snout or great mandibles (your choice). While you are raging, as a bonus action you can make a special attack with your bite. If the attack hits, it deals 1d8 piercing damage and you regain a number of hit points equal to half your barbarian level (rounded down) + your Constitution modifier (minimum of 1 hit point).
• Claws. Your hands transform into claws, which deal 1d6 slashing damage on a hit. While you have two hands free, when you take the Attack action on your turn you can make one additional attack using your claws as part of the same action.

When you score a critical hit with a natural melee weapon attack, you can roll that weapon’s damage dice one additional time and add it to the extra damage of the critical hit.

I think you're undervaluing the Bite attack a bit. Should you choose the bite option when you rage you have access to both hands for a two handed weapon or sword and board style combo. Like you mention, this option lets you maintain a higher damage option 2d6/1d12/1d8+shield for one of your attacks while keeping a healing option at the ready should you need it. 

Also with the Tail attack you don't need any hands for that weapon so it is a 1d12 weapon that you can use a shield with. Its not the most damaging option out there, but it has a place in some more defense oriented builds that want to keep enemies at bay. 

I think having a weapon would be better than using the Claws and that the attacks should be a Bonus action. Here's how I would make this ability.

Form of the Beast 3rd-level Path of the Beast feature

When you enter your rage, you can transform, revealing the bestial power within you. Until your rage ends, you manifest a natural melee weapon which you can use a Bonus Action to make 1 attack with, choosing one of the following options each time you rage:

• Bite. Your mouth transforms into a bestial snout or great mandibles (your choice). Your bite deals 1d8 piercing damage on a hit. Once on each of your turns when you damage a creature with your bite, you regain a number of hit points equal to your Constitution modifier (minimum of 1 hit point).
• Horns. You grow horns out of your head, which deal 1d8 piercing damage on a hit. Once on each of your turns when you damage a creature with your horns, the target must succeed on a Strength saving throw (DC equal to 8 + your Constitution modifier + your proficiency bonus) or be knocked prone.
• Tail. You grow a lashing, spiny tail, which deals 1d10 piercing damage on a hit and has the reach property.

Three varieties of bonus attack! I like it.

To get the 'most' out your feedback go onto the WOTC site n actually fill out the feedback form

Bite isn't meant to compete with 2 handed weapons. it's a healing/damage dealing thing. You can use 2 light weapons or a 2 handed weapon and still have the bite. hell you could use a halberd and get PAM to do a d8(that gives you 5 HP), a d10, and a d4.  also this works with the tail as well. the tail is the only d12 with reach in the game and it leaves both hands free. so you can use a shield or two and still have a d12 weapon with reach. 

in short: the subclass is fine as written. 

Horns/Gore should be a weapon option for this subclass, but I still like claws. I do like this subclass, and don't think it needs any fixing, besides some possible added options.

Don’t get rid of the claws, this is Wolverine!

Ok, but I still think the horns are necessary.

I would like for barbarian path of the beast to be able to take the 3 shapes at the end, I think it would be much cooler to play with the class if it were like that, but I also know that this would bring several balancing problems if you took more of 1, so I suggest slightly nerfing the damage and healing of the class and allowing that at the end or just before you get the 3 shapes, bite, claws and tail, follow the suggested changes for this:
bite: fixed damage and healing.
claws: unarmed attacks deal 1d4 of additional damage, you can still make an additional attack with them if you attack with them.
tail: damage reduced to 1d8

Does the fact that the claws are "natural weapons" make them available for TWF?

Quote from ArChelios >>

Does the fact that the claws are "natural weapons" make them available for TWF?

 No. This was answered on Twitter by Dan Dillon. https://www.sageadvice.eu/2020/01/15/barbarian-path-of-the-beast-claws-since-it-doesnt-have-the-light-property-i-cant-two-weapon-fighting-correct/

My one annoyance is that the flavoring gets kind of wonky. Like, if you're playing a character descended from a werewolf, it would be really stupid to use the tail option, but it also does better damage at low levels. Similarly, if you're playing someone connected to a crocodile, only swim speed would really make sense. There's ways around it, but the flavoring can still get kind of weird. I'd say to change it to a few sub-paths like Totem Warrior, but that would kind of kill the appeal of the subclass. 

We just changed the damage to bludgeoning for the tail. 

Has anyone made any good homebrew 14th level options for this? because the one as written is pretty silly. 

I think even though it's not a beast, playing as a dragon descendent barb might be fun XD They have all bite, claws and tail, so it might make sense thematically. Plus you can RP the rage as though going into a draconic rage and f-ing shit up XD

The tail also lacks the heavy property, allowing small creatures to use it without disadvantage.

Additionally, it is the only reach weapon other than the whip and a mounted lance that is used without two hands. I am not certain but would this (as long as you are not holding another reach weapon) allow you to have two ranges to provoke opportunity attack at.

Me and my brother (who is playing DM and a character) decides to let me run into a room alone and waits with our sidekick outside, while I get ear ****ed by a banshee just so I pass out and he runs in saying his form of the beast at level 3 gives him magic claws so it does full damage. Please tell me that these aren't magical so I can shove it in his face. (This isnt the first time he's bent the rules out of my favor and into his saying "I'm DM so I make the rules)

Level 3 in the UA no, its lvl 6 that the claws become magic

They are definitely not magical at level three, using TCoE.