I think you're missing a stronger version of what you said: a "bad" mathemtician might not be able to find a proof (or an elegant proof, in the case of already proven theorems) but he or she will seldom find an incorrect proof (or a fundamentally incorrect proof – there can be subtle errors). This is because of a fundamental property of proofs: verifying a proof is, in theory, a mechanical process. Of course, in real life it isn't, but it is still possible to apply rigor to ascertain, with almost certainty that either you proved something or that you find a step in the proof which is "not obvious". Of course, it may be not obvious because one is too stupid to see why it is true, but at least you can come up with a concrete problem with the proof.

The other point is historical, rather than technical. Mathematicians often found shorter, more elegant proofs, but in most cases these were people who did understand the original proof well enough to modify it, not someone coming out of left-field with a whole new theory – and when it was (non-standard analysis proving results in functional analysis is a good example), it was not rejected by the establishment as correct [though it might be regarded as useless]. This should show evidence that the cranks, and not the mythical "mathematical establishment" is at fault :)