This argument is seemingly related to Gregory Chaitin's views on noncomputability and nondefinability. Chaitin observes that almost all real numbers cannot be referred to, or singled out, by any mathematical method available to us -- for example because definable numbers using a language or notation must have the cardinality of the natural numbers, but we know from Cantor that the cardinality of the reals should be larger.

Chaitin simply thought this was an impressive fact about the reals and the limitations of mathematics -- a way in which mathematics contains randomness and that many or most facts are "true for no particular reason". This author instead seems to conclude for a related reason that the reals don't exist because we have (and could have) no usable technique to distinguish most real numbers from one another. His complaint in this video is a Chaitin-like observation that we have no way to distinguish real number A from real number B in a finite amount of time or with a finite amount of reasoning or information, and an un-Chaitin-like conclusion that maybe we then have no reason to believe that these numbers exist and are distinct from each other.

Edit: and he emphasizes later that if we believe in the reals, numbers must exist that we can't actually do arithmetic with (which I would suggest is sometimes for the Chaitinesque reason that we can't name or define them, or other times for the weaker Chaitinesque reason that we can't calculate their values), so he seems to ask what good such numbers are to us or what reason we could have to believe that they are real.

(For anyone else reading this who might be misled, note that OP's position is generally considered to be "crank" mathematics)

Uhhhh...no. Take a look at Dedekind cuts.