> and they aren’t irrational (i.e. they have a finite precision).
I'm not sure if I'm misunderstanding what you mean by 'finite precision' but the ordinary meaning of those words would seem to limit it to rational numbers?
In practice you're always computing with finite precision. (Even computing with symbolic expressions is just a preliminary step to what's ultimately a numerical result with finite precision.) The whole point of real numbers is to abstract away from detailed considerations of precision, and figure out what happens if you only ever care about putting satisfactory bounds on the output and are willing to bound your input to the extent required.
The idea of R is that it allows you to reason about things like "I need more than X input precision to achieve Y bound on my output". Just sticking with naïve computation in Q does not suffice for that.
I'm not sure if I'm misunderstanding what you mean by 'finite precision' but the ordinary meaning of those words would seem to limit it to rational numbers?