I think Norm Wildberger's videos are very useful to think about and enjoy though certainly one should reflect carefully on what he says. His videos inspired my work on a new definition of real numbers. The basic idea is that it is the set of all intervals that contain the real number. Since this is circular, there are properties that describe when a set of rational intervals are defined. This approach is equivalent to Dedekind cuts and does not address Wildberger's concern.
But there is another definition which I call oracles which does a better job. It is much more constructive and is about a a procedure that one can ask whether a fuzzy version of a given interval contains the real number. It has various properties for a procedure to satisfy and, if so, then it will generate the set of intervals that contain the real number if taken out to infinite length.
So basically, it is a two part-definition. There is a theoretically perfect version and then there is another that yields to the practical problems of not being able to actually specify a real number entirely.
If interested, the papers are hosted on GitHub [1]. The most recent version going over what I just said is Real Numbers As Rational Betweenness Relations [2]
2: https://github.com/jostylr/Reals-as-Oracles/
1: https://github.com/jostylr/Reals-as-Oracles/blob/a98472813e9...