Forget talking about "bigger", "larger", "more", "size", etc when talking about infinite sets because these concepts don't really make sense for infinite sets.
This seems to be the key thing, especially after checking out some of your earlier links on cardinality and such. I can happily accept the proofs in the videos if we're talking about concepts that are specific to infinity, but saying that this means (for example) there are more numbers between blah than there are blah, doesn't seem to be a simplification or a translation to layman's terms, it just seems to be incorrect.