those who see continuous distributions as just an abstraction of a world that is inherently finite vs those who see finite steps as the approximation of an inherently continuous and infinitely divisible reality.

How about neither? Math is a formal system (like a game). It has no inherent relationship to "reality" or physics. There are only a few small areas of math that have been convincingly used in physical models, while the vast majority of mathematics is completely unrelated and even counter to physical assumptions (eg tarski paradox). Questions about the finiteness or divisibility of "reality" are scientific, not mathematical. Etc.