Again, I don't think your position is indefensible, but this doesn't strike me as particularly convincing. The usual definition of R is that there exists a unique ordered complete Archimedean field up to isomorphism. We get the kitchen sink from the least upper bound property. As a constructivist you're gonna say that I don't get to define R like that, but you can't pretend it's done for no reason or that it buys nothing.
> I certainly learned about areas in geometry
And how were they defined? In elementary geometry we just sweep the question under the rug, usually...
If you get to say that being able to articulate why the measure of Q is 0 is unimportant and uninteresting, then I get to claim that the supposed problems with the usual definitions are also unimportant!
Saying that the non-constructive world leads to worse problems is a respectable position. Pretending the usual way of doing things is completely arbitrary isn't very honest.