Assumptions are assumed, not proven; there is no such thing as an unproven assumption, only unjustified ones.
Any assumption that is unproven is an unproven assumption. Of course they exist.
Now, "unproven assumption" isn't an error in itself, but in this case it is where the error lies. This particular "proof" holds only for cases where the assumption is valid, and the conclusion doesn't follow from from the "proof" until this assumption is proven to be valid.
That's just not so, though we are well beyond the OP's original question by now. Induction proofs follow a particular mechanic. They follow only
that mechanic. To hold them to a nonexistent stricter standard is itself an error, and raising it invalidates your criticism of the proof.
In the step of an induction, you don't prove
-- you don't even question
whether the proposition is true for K. The only thing that matters
is "if it is true for K, is it true for K + 1?" (in "strong induction", read "if it is true for [N0 .. K], is it true for K + 1")
But if there is some K for which the step fails, then the proof fails. The only
K for which the strong inductive "proof" of "all whole numbers are even" fails is K = 0. If it were not for that single case, the "proof" would be correct as written.
In real, important, and interesting problems, the importance of proof by induction is that you never
have to prove or even justify
the assumed side of the step. This allows you to prove many things that otherwise would be tedious beyond all practicality.