I paraphrase with italics from the original post to give my perspective:
My naïve expectation would be that metamathematicians * might take a difficult construction or proof, and clarify it by isolating the key ideas behind it. Having isolated the key ideas, they might then highlight their relevance and thus point the way forward. Beyond this, I would hope that such practitioners might elucidate the `true meaning' of axioms and of definitions by examining their ontology in a wider context.
I first took meta-mathematics hoping to discuss things like the duality between the notions of points and lines in axiomatic geometry, so that one could find ways to discover and prove more results by exploiting the symmetry of the foundations of a given theory. I found something completely different, of course, with only occasional references to duality or n-ality. I put * after metamathematicians, because, after some years seeing how some good mathematicians practiced (they would not stop at a single result: they would poke it, prod it, extend it this way or that, try to find limits by tweaking some aspect of the assumptions or proof and come up with counterexamples, and use other methods to look for wideness of application, or anticipate when such attempts might fail), I realize now that the first term should be "professional mathematicians", or just "mathematicians" perhaps.
(This is not to dismiss philosophy from the question, so much as to ask for an alternate basis for the question. I think philosophical examination might yield guidance as to which logical system to use in certain pursuits, but it would still depend on the naive expectation above being carried out by mathematicians.)
Gerhard "Mathematicians Know What 'Is' Is" Paseman, 2014.10.01