Here are some (somewhat disconnected) thoughts.
The main sticking point expressed in the objections is over the word 'breakthrough' (which is indeed difficult to characterize). But the Calabi conjecture example, and Sam Hopkins' answer, suggests a slightly different sort of question which might (might) gain more traction. "What are some examples of theorems which were eventually proven after initial attempts to prove them wrong?"
It's probably not a great question posed this way, but the situation I have in mind is familiar in research: we have a statement whose truth we are unsure about, but we have our instincts about which way it goes, and this initiates a kind of proofs-and-refutations dialectic where our initial hunches and approaches morph into their opposite, e.g., efforts to prove something is false eventually yield insights into the fact that the statement is actually true. Or something like that. Anyway, I think some such question might have appeal, and would highlight one of those "nontrivial twists" (as you put it) that helps make mathematics interesting -- and it might also be free of the vagueness or opinion-based aspects that people are objecting to.
About those other three questions: keep in mind that MO 'culture' is always evolving, and it's not necessarily the case that a question that was popular in 2010 would be popular (or even remain open) in 2014. I can see an objection that the first of those is opinion-based (what does 'surprising' mean, etc.). The second strikes me as less opinion-based somehow -- but I was personally disappointed by the quality of many of the answers. The third was not at all opinion-based or too broad: it was quite concrete, and unobjectionable IMO.
Comparing the merits of these sorts of soft questions (yours included) is largely a matter of taste; I have just given you my own personal reactions. It would probably be best not to try to argue matters of taste (e.g., which is "more serious" than the others). It would also be best not to press the idea that there is some sort of bias/discrimination against you personally; it would be impossible to prove, and such discussion would be headed in an ugly and fruitless direction. It is however true that there is a built-in looking askance and being more strict with 'soft' questions, since most users don't want MO to turn into a repository of subjective opinions, gossip, etc. (this speaks more to the broken windows metaphor).