Huh? There are no counterexamples to the Euler characteristic of a polyhedra being 2, and the theorem has generalized beautifully. If anything conditions have been loosened as new versions of the theorem have been used in more places.

Well, what do you mean by polyhedron? Consider for example a cubic nut. Does this fit your intuition of a polyhedron? Well, since it has genus that is not equal to 1, it doesn't have Euler characteristic 2. And the original proof that V+F-E=2 didn't handle this sort of case. (That's one reason why people often add convex as a condition, to deal with just this situation even though convex is in many respects stronger than what one needs.) Cauchy's 1811 proof suffers from this problem as do some of the other early proofs (although his is repairable if one is careful). There are also other subtle issues that can go wrong and in fact do go wrong in a lot of the historical versions. Lakatos's book "Proofs and Refutations" discusses this albeit in an essentially ahistorical fashion.