The appearance of new traits is already a given, as per your words:
You’re saying that UCD with variation from mutations can do that once, but never twice. Why not?
Your sense is wrong. You make it sounds as though we were all happy, going along with our nice neat story of evolution and inheritance and then suddenly someone noticed this horrible analogy, and everyone scrambled to give it a name to make it compatible with the existing theory. Convergence has been known about from the very beginning of the theory of evolution - it’s kind of hard not to notice that both bats and birds have wings, for example. The vast majority of the time convergence is actually easy to spot because it actually doesn’t give the impression of common ancestry. Take bird and bat wings again, for example. It’s immediately apparent that while they’re (very superficially) similar, they’re actually very different. This is because convergence (by definition) starts from at least somewhat different starting points, and because there is a major element of chance in variation, and because selection pressures can be quite different. Birds and bats superfically might be mistaken as close relatives because of their wings, but it doesn’t take much effort to realise that they’re very different. Ergo, their wings evolved independently. If the animal kingdom was just replete with examples of “perfect convergence”, more closely resembling pathworks of different specific morphological characters with no discernable pattern (like horses with shark teeth, primates with bird wings, fish with kangaroo pouches), you might be onto something. But the world doesn’t look like that at all, does it?