No, the rationals are not discrete in the usual topology. They end up being discrete when we consider continuous mappings from R->Q though. That is the "technical" sense that I refer to. The rationals, as you say, are dense in R but they are also dense in the computables.

The big Cantorian leap that we make is when we use the diagonal argument to argue that the rationals are countable. All the real construction techniques (Dedekind cuts or Cauchy sequences) effectively only yield the computable numbers, the real numbers outside of the computables are inherited from the diagonal argument rather than being foundational to the construction. I mean, this is trivially true because constructions are constructive.

I disagree that area is tethered to measure theory; I certainly learned about areas in geometry long before I ever heard of anything with measure theory. Measure theory exists to tie up some of the horrifying poorly behaved functions that increasingly wily mathematicians invented to break our notions of area and continuity. But we have better tools now for dealing with those that don't involve measure theory so there's no reason to ever hear the phrase "almost everywhere" or "subadditive" ever again.

To back it up to your closing and my main point -- the constructive numbers are way closer to the way we work with numbers because all numbers we ever deal with, even abstractly, fit this definition much better.