This isn't a question of the weapon quality vs unarmed, but rather, what would the consequences be for a GM to rule that they could be used as dexterity weapons rather than strength?

To me, they should be much more dextrous than even a dagger, so why not? (on a realism level)

However, would this "break" anything? Given that these are 1d4 attacks, I'm thinking not, but I invite commentary.

PS - I had visions of a Vashino rogue which uses it's claws, teeth, and tail instead of daggers.

Two reasons. First, every finesse weapon except the whip is a piercing weapon and Tabaxi’s claws are slashing weapons, so they don’t match the damage type of weapons that use dexterity for attack and damage bonuses. And second, Tabaxi get a +2 to their dexterity and no bonus to their strength so making their claws a finesse weapon gives them a racial advantage to hit and damage on top of their racial advantage of 1d4 damage instead of a flat 1 damage. 

Its subtle, but it is a second bonus on top of the first one that unbalances the racial abilities. 

Thanks for the feedback. I don't think the damage type matters much at all in 5e. Vashino also get a +2 dex, Tortles on the other hand get a +2 strength, thereby having an "unbalanced" racial.  Therefore, IMO the differences you listed aren't really a breaking issue (for me).

Any other possible reasons?

I think it works best with strength dex is already a great stat it doesn't need to be even better then it is. 

There reason it's tied to strength is because the unarmed attacks of every other PC are also tied to strength, just only doing 1 damage. If you want it to use Dex, do what any character can do to make their unarmed attacks use Dex, take a level of monk.  

I would also allow a feat that gave a +1 to Dex and let you use Dex for unarmed attacks

Quote from Tim >>

First, every finesse weapon except the whip is a piercing weapon

 The scimitar is also finesse.

Of the six finesse weapons, two are slashing (whip, scimitar) and the other four as piercing. None are bludgeoning (the terms "finesse" and "blunt object" do not go together).

Interestingly, only two of them (dagger, dart) are simple weapons; the rest are martial.

As an example of what you're talking about, there's the case of the Blood Hunter Order of the Lycan. Attacking with the claws uses STR or DEX bonus and it is spelled out specifically in the feature because it is an exception to the rule. This is also a class as opposed to a race in the example you gave, but it's the closest approximation I could think of.

As a house rule, I'd say it doesn't break anything mechanically with the game.