I don't find such a category [the 'non-contingent'] useful because b) of all the instances where philosophers had to revise their first-principles derivations based on subtle assumptions about how the universe works.

This sounds like a metaphysics-epistemology confusion (or 'territory-map confusion', as folks around here might call it). It's true that empirical information can cause us to revise our 'a priori' beliefs. (Most obviously, looking at reality can be a useful corrective for failures of imagination.) But it doesn't follow that the propositions themselves are contingent.

Indeed, it's easy to prove that there are necessary truths: just conditionalize out the contingencies, until you reach bedrock. That is, take some contingent truth P, and some complete description of C of the circumstances in which P would be true. Then the conditional "if C then P" is itself non-contingent.