You can't define definability, because otherwise you could talk about the "least natural number not definable in fewer than n symbols" which is a contradiction for sufficiently large n.

Doesn’t definability just depend on the system you’re using to generate numbers being able to represent it?

For any system we could use to define a real number, (and not just because we are mortal), the system could only define or uniquely specify countably many real numbers. Any countable set of real numbers has measure 0, and therefore, almost all real numbers cannot be defined/specified using that system.

This holds for whatever system you use which defines a (possibly partial) function from [ the set of finite (but unboundedly long) sequences of characters over a finite alphabet], to real numbers

Or equivalently, from the natural numbers to the real numbers.