I just don't see any reason for Avatar/Robin to be playable. I know he/she's important in Awakening, but Chrom is technically still considered the main character (he's on the cover art, promotional materials, crossovers, et cetera). And in Smash Bros., the only one-off characters that are playable are the main protagonists. That's excluding the fact that the character is a customizable avatar -- and don't compare that to Animal Crossing, since you play as a villager in every game in the series.
I wouldn't count him out. He's clearly the most unique, yet still important potential Fire Emblem newcomer. He would also represent the Tactician/Avatar character, which has appeared in three games now (but fleshed out the most in Awakening). And it's this tactician who actually commands the characters within the game and devises the strategies. It's probably the best way to represent the essence of Fire Emblem, especially since Robin is capable of both wielding weapons and using magic, which are both staples of the series, not just swords as seen with Marth, Roy, and Ike. He's pretty much the main character along with Chrom, anyway. I'd even argue that's he
more of a main character than Chrom in pretty much every aspect except marketing, because he is the character that YOU take the role of, and he's arguably more important to the plot, too.
He probably doesn't appear in much promotional material because of his customizable nature and due to the fact that he's supposed to be YOU, but that doesn't mean he can't be a playable character in Smash Bros. He has defaults. They put Villager in the game with his defaults, too. I know you said not to compare it to Animal Crossing, but why not? Yeah, Villager makes sense because you take the role of that character in every Animal Crossing game, but Fire Emblem isn't really like that. Fire Emblem is pretty much made up entirely of one-off characters, and Robin is arguably a/the main character of Awakening, and he represents the franchise better than anyone else from the game, which is important. I can imagine Sakurai choosing Robin for the same reasons he chose Pokémon Trainer.
Chrom wouldn't really add anything new. I mean, sure, they could always make a different move set for him, but I can't imagine him being much different from Marth, and I don't think he's what the Fire Emblem franchise deserves for a third playable character in Smash Bros. We need a character that can better represent the essence of Fire Emblem, and Robin is that character.