I voted yes.
Mutant horses would be awesome.
They could be skeletonlike with fangs, spikes and bone armour.
Having a mutant horse as a companion would be cool.
If they died you could get a replacment from their herd or their family. Like when Dogmeat dies you get Dogmeats Puppy.
You could also have mutated ponies.
If there were a lot of them you could get a perk from a quest to summon them if one dies like in Red Dead Redemption.
As to extinction.
I suspect the exstinction explantation has been applied to a few animals because they could not or did not put them in the original games. A need to explain everything too early in a too nialistic way.
I don't think wild horses are as fragile as domesticated ones.
And they would be wild by 2077 or later.
And mutated to some degree.
There are lots of herds of wild horses allover the place. They are shorter and tougher than domesticated ones I think.
Domesticated horses are bred to silly and cruel specifications sometimes I think.
Race horses are bred to have thin bones and be light and thin.
Not all horses are that fragile I think.