Maybe I'm getting hung up on words, but my beef is with the parent saying they find real numbers "completely natural".

It's a reasonable assumption that the universe is computable. Most reals aren't, which essentially puts them out of reach - not just in physical terms, but conceptually. If so, I struggle to see the concept as particularly "natural".

We could argue that computable numbers are natural, and that the rest of reals is just some sort of a fever dream.