This may just be totally ad hominem speculation (though there's no particular person I have in mind), but...
Maybe the reason why some are inclined to think we should always be able (in principle) to reflect and formulate a system that captures any recursively reasoning we do about math and logic has to do with a certain view about reasoning + rule following.
Remember how Wittgenstein (in the Blue and Brown books) criticizes this theory that we manage to obey the command "bring me a red ball" by first imagining a red patch to get an idea of what color "red" is, and then picking the color that matches this imaginary patch. He says: do you first need to imagine another red patch, in order to know what color to make your sample patch? So, this theory leads to a kind of regress. [I'm always tempted to call it the smoke-two-joints-before-you-smoke-two-joints theory of understanding language, after Bob Marely's famous song about regress]
Presumably everyone will agree that in this case we need to just posit (and perhaps scientifically explain) a direct ability to pick red things, when commanded to. Invoking a further layer of person-level thought (where you pick the red balls by first doing something else like imagining a red patch) just leads to regress.
But, an analogous theory with regard to mathematical reasoning would be that when we are asked to answer some mathematical question, what we do is first consider certain rules for how to reason about mathematics, and then do what these rules say. Now, I think this is a very bad theory. But, if you accepted it, you might think there could well be a special process of reflection where you, in effect, remember these rules, or become consciously aware of the rules you were unconsciously appealing to all along. That is, you might think: any recursively enumerable portion of mathematical reasoning you accept, you should be able to formalize (and recognize to be correct) by making explicit all of the (presumably finite number of) rules that you implicitly consulted when doing that reasoning.