In general he thinks a lot about tool kits and I think what we're seeing here is the math equivalent to being great at debugging.
Exactly what I say at parent-teacher conferences.
Nelson was making an extraordinary claim so the odds were definitely against him.
The latter implies 'ooh, he won;' the former only 'ah, there was a problem and he spotted it.'
The useful skill, here, is one shared by logicians and compilers: the ability to quickly find flaws that invalidate a proof, without needing to fully understand what the proof is trying to prove.
Said skill, if it could be more widely-learned, would be very helpful in iterating toward sound proofs (or impossibility proofs for such.)
But when it's only a skill in the hands of another party external to the one writing the proof, that iterative aspect isn't really there.
I think I agree with the rest of what you have said.
Invalidating a proof is essentially what a compiler does when it blows up over a typo in the code you've fed it.
The thing is, most of the time, it is just a typo. The code as-is is wrong, but the code with one character different — the code if "branched" in an alternate direction for just a few millimeters, and then course-corrected back onto the original existing path — is right, and the semantic meaning of the code isn't changed/compromised by that correction.
Finding a logic-level flaw in a (presented) mathematical proof is usually similar: 99% of the time, it's something fixable.
That 1% of the time does still exist; there can be "fatal flaws" in proofs, where the proof's author can't find a way to salvage their proof. But it's not the invalidation of the particular proof, where the knowledge that the proof is unsalvageable (i.e. that there's no path that follows the general "plan" of the branch to get from A to B) gets created. That knowledge is only discovered after much more work by the proof's author, to try to "dig around" the problem, that all turns out to hit other walls.
(Note here that I'm assuming that the proof isn't already believed to be true when it's invalidated. Which it usually isn't; most proofs that get invalidated are invalidated when they're still being circulated as something novel and for-scrutiny, rather than when they're already generally-accepted. If a proof was already thought to be true, then invalidating it would temporarily create "negative" knowledge — it would retract a previous consensus assertion of the truth-value of the proof.)
Perhaps the most effective thing a person can spend their time on is pointing people along paths likely to lead to success, and guiding them away from dead ends.