[lmkg]

The Reals are venerated because they're actually usable. Other numbers systems tend to be a gigantic pain in the ass to get any work done with.

The Reals are constructed specifically to be the smallest set that has some nice algebraic properties, like Least Upper Bounds. Sets that model the real world, like the constructables, countables, computables, etc. tend to be subsets of the Reals, and therefore don't have those properties. That absence makes life difficult.

The Real Number system, like almost everything in mathematics, is an approximation of reality that makes a trade-off between faithfulness and tractibility. As it turns out, gaining more of the former loses you quite a bit of the latter. It's generally not worth it.