You must not be familiar with his filmography. He has played though characters; he plays a FBI agent Joesph D. Pistone in "Donnie Brasco" who went undercover to infiltrate the mob. In "Nick of Time" he plays a mild mannered accountant. In "From Hell" he plays a police officer who investigate the 'Jack the Ripper' murders. In "The Ninth Gate" he plays a rare book dealer. And of course, in his first major role, he plays an undercover police officer in "21 Jump Street".
As for his look, I wouldn't say he looks "all American surfer" as that would suggest blonde hair and blue eyes, whereas Depp has olive skin, dark hair and eyes. Because of his Native American (Cherokee) heritage, his 'look' can be manipulated in a way that he's been able to play more than one ethnicity (Italian American, Cuban, Spaniard, Gypsy and rightfully Native American) convincingly. But he does look like a rock musician, as he is one before he became an actor (I don't know if you knew that).
Although I have to agree, unfortunately, that the general public and his rabid fangirls could ruin the movie's marketing and overshadow the film. Which is highly unfortunate, as I'd love to see him in Nolah's Batman. Perhaps in a smaller role (like a cameo) as a lesser known character, would work. Also, a drastic change from his normal look (long hair) should also do the trick. Cut all of his hair. Hell shave it, like he has it here.