It has always bugged me that people call the numbers that Cantor 'constructs' the real numbers.
Underlying it is the axiom of choice, which says that it is meaningful to talk about the results of processes that would take an infinite amount of time to complete. (e.g. choose an infinite number of digits that create a unique number.)
A number like 0 or 75 or 1/3 has a name, in fact all the integers and rationals have names. You can pick one out individually and talk about it. You might encounter it in some way.
Some irrationals (sqrt 11) and transcendentals (4 pi) have names. Most don't, and you will never see them as individual numbers, only as members of an interval.
However, I agree with the general thrust of your argument: it seems weird, in hindsight, to call them the "real" numbers since most of them do not meaningfully correspond to anything in physical reality. Especially since most of them are non-computable; i.e. there does not exist any computer program that outputs their digits in order. It seems difficult to say anything meaningful about those at all!
In case you haven't seen it, the proof of this is simple: there are only countably many computer programs, and uncountably many real numbers. Thus almost all real numbers are non-computable.
In fact I am not convinced that there is anything in physical reality that meaningfully corresponds to an "uncountable set" in mathematics.
The real numbers are useful because their behavior as a set matches our intuition about how numbers should behave, even though most individual members don't. I believe this is what you are alluding to in your last paragraph.
But anyway, they were named "real numbers" in opposition to "imaginary numbers", not rationals or computables or anything like that. Which is even more unfortunate because the construction of complex numbers from real numbers is extremely straightforward and not mysterious at all (just define them as pairs of reals with a particular multiplication operation).
Underlying it is the axiom of choice, which says that it is meaningful to talk about the results of processes that would take an infinite amount of time to complete. (e.g. choose an infinite number of digits that create a unique number.)
A number like 0 or 75 or 1/3 has a name, in fact all the integers and rationals have names. You can pick one out individually and talk about it. You might encounter it in some way.
Some irrationals (sqrt 11) and transcendentals (4 pi) have names. Most don't, and you will never see them as individual numbers, only as members of an interval.