I don't get your logic... so you are saying the Pokémon Trainer in Brawl is really someone else who is not Red... maybe he is Red's cousin Crimson... just because Pokémon Trainer has no name in Brawl?
Pokémon Trainer + Red's appearance design + no name = a person who is not named Red but someone you get to name
So... why can't we just name him and say he is Red?
Because he isn't Red.
And your second paragraph is 100% what he is, except for the fact you can't legitimately directly name him in Brawl anymore you can name regular Samus in Brawl.
Basically, any trainer name you give him is not official and, get this? It never will be. If people want to call him a false name, I'll stick with a better one called Crimson. It's just as accurate anyway.
Clearly you didn't get my logic if you think I actually think he's Red's cousin. What my logic is is that anecdotal evidence is nothing but pure crap and that's it. You can call him as you want for a fun nickname. But never ever treated him being Red as a fact, because, you know, it quite clearly isn't. This is already proven by the fact that every official(and not fanmade, since those are stupid to actually accept as a fact) source absolutely refer to him as Pokemon Trainer. Also, his Trophy describes him as the class entirely. That pretty much proved that he just isn't Red. If he was Red, at least his Trophy would, you know, refer to what he did.
Appearances are not actual characters at all. Names are when it comes to Smash if you want to be accurate.