I don't think the "America's Team" stuff even matters as far as why we get so much attention. If a Yankee or Laker says something even remotely controversial, it's going to get just as much attention as if a Dallas Cowboy says it. Nobody cares if an Astro or Timberwolves player says it, same as in the NFL. Our team just happens to be the one that's historically successful and therefore has tons of fans (and haters).
I do think we're a little unique in that we happen to play in the East against several of the major population centers in the country and for decades we kicked their teams' asses, so lots and lots of people hate us. And of course the media plays that up because it gets them ratings.