That's certainly true, but that's also clearly not what was being talked about. The problem was (easier) avoiding choosing bases to perform a construction, and

That's certainly true, but that's also clearly not what was being talked about. The problem was (easier) avoiding choosing bases to perform a construction, and (harder) avoiding assuming all vector spaces have bases. Using bases that you directly contruct isn't something anyone ever really has reason to avoid. (And really, neither is the second in the finite-dimensional case, but hey, may as well if you can, right?)