Who says the question was to demonstrate an inductive proof? Being able to prove that all horses are the same color if all pairs of horses are the same color isn't particularly useful, as typically that would be the definition of all horses having the same color. All that humbug didn't add any extra information to the problem at all, it didn't make things clearer it just made them harder to understand.

Right, the goal here is to find the logical error in an obviously wrong proof, not how to fix this proof as the proof is obviously wrong from the start. You trying to argue how to "fix" it is the one side-tracking the whole thing.

If you say "B wouldn't be wrong if you changed A, so A is wrong!" then that isn't how you find errors. If A is a correct statement and B is an incorrect statement then B is the wrong statement. If you can make B correct by changing A, then B is still an incorrect statement.

If this was a memorization question where you were supposed to find a particular A and B, then you could say that A is wrong even if the statement is correct, since you knew what A and B are supposed to be. But it isn't, we are just here to find the error in the statement to prove that the proof is wrong, we aren't here trying to find a correct version of this statement.