The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.
This is a problem of modeling optimization. The models based on uncountable "real" numbers are logically consistent and simple to use, so they are adequate for predicting what happens in natural or artificial systems.
All attempts to avoid the uncountable infinities produce models that are both more complicated and also incomplete, as they do not cover all the applications of traditional infinitesimal calculus, topology and geometry.
Unless someone will succeed to present a theory that avoids uncountable infinities while being as simple as the classic theory and being applicable to all the former uses, I see such attempts as interesting, but totally impractical.
The real numbers are a useful mathematical trick that make it possible to prove results in calculus. What you surrender in return for being able to prove statements is to give up the ability to compute expressions. This may be a worthwhile trade-off for physicists but for the universe (which does many computations and zero proofs) it's quite a burden.