Maybe Mario would if he reincarnated in another time every game
But Link's reincarnation is nothing more than a byproduct of Nintendo not wanting to link up the storylines of every single Zelda game. The only reason they set things 100 years in the past or in the future is to obviate the need for a coherent, continual story. The 'different' Links across various games are not because Nintendo views Link as more avatar than character.
And I wonder, have they ever spoke of Link as an avatar outside the context of not adding voice acting to the game? I seem to remember 'link as an avatar' coming up for the first time only then, at least from the Nintendo guys like Miyamoto and Anouma.
I'd be curious if there are comments predating WW or TP (the time when I think people started looking for voice acting to be added to Zelda) describing Link as an avatar.
In any case, I agree with the others who view Link as more of a character than just an avatar. There's over 25 years of history now of Link appearing the same way, not only in the mainline Zelda series but in others as well. It seems arbitrary to me to all of a sudden make him into a girl.
Having said all that, the avatar argument isn't entirely without merit. I think a lot of the desire for Link's gender to be selectable probably stems from the role playing part of the game, where you do become a hero in a simple story to rid the land of evil, and people want to choose the essential elements of the character they're going to use to do that. I'm sympathetic to that, so on that basis I wouldn't object to full character customization, as long as the official Link remained the same character he always appeared to be.