Happening, selecting, picking are either physics, or computation. To paraphrase Yoda, "The real number just is, there is no picking".
My problem is not with infinite digits. You cannot actually pick a random real (or at least ever finish picking it.) This business of saying "it happens all the time" is pure H.S. That is neither logical, or mathematical. For an example that is easier to unwind, look up the formal definition of the axiom of choice. There is no choosing involved.
The metaphors of choosing, or "picking a random real" are crutches, much like those pictures you are always railing against. I think those metaphors though, are more insidious than pictures, because while its easy for a mathematically mature person to distinguish between a picture and the mathematics that may map to that picture, the metaphors surrounding probability are more ingrained and unconscious.