> absolutely nothing requires an incomputable continuum of reals.
In a sense this is wrong. Some say the indispensability of modern math, which includes the incomputable reals, shows that such abstract objects are required for science.
The argument is more like our most successful sciences like physics use our most successful math, which is ZFC. And thus incompatible reals are part of this package deal. Maybe it’s possible to do physics without any packaging of lots of these mathematical objects, but to my knowledge it hasn’t been done (physics based on a paired down math, from top to bottom). It’s a de facto require, that despite effort like intuitionism or Science without Numbers has not replaced classic ZFC as our most successful math.
If a popularity is the measure of success, then ZFC is successful. However, I'm not sure there's anything that requires it and doubt that there's a strong claim wrt to any ZFC requirement. So saying science requires it, is like saying that ARM is required vs x86 or IEEE 754 is because your experimental setup runs on it.
In what way does analysis require incomputable reals?