My mother would be appalled. America HAS Talent. Would you say "America Has Got Talent?" No, but that's what you say with the apostrophe. I wonder why my fourth grade students can't figure out how to write a paragraph...

Love the show. Hate the title.

America's Got Talent is also correct, although not the way most people would speak but it is still correct. Even if it wasn't, it would still be okay to use as a title because then they are just taking artistic/creative freedom.