Before we get started, let me say this: I like Bob Bradley. I like him a lot and think he was criminally underrated as United States manager.
That said, what he said about Michael Bradley and not being signed by Arsenal is ludicrous. It reeks of a father getting protective of his kid, which ... wait a minute, Michael is Bob's son.
"I think American players and coaches have to fight really hard for respect," Bob Bradley said to Slate. "In January, Arsenal [was] looking to add a midfielder, and they chose Kim Källström. Kim Källström’s not a bad player, but I think Michael feels pretty strongly that he’s better, and so Arsène Wenger must not feel that way, and [Arsenal chief executive] Ivan Gazidis must not feel that way. So sometimes, no matter what you do, you don’t get the respect you think you deserve."
Well yes, Michael is better than Källström, but there is also something relevant here -- Källström went to Arsenal on a loan and Michael would have cost £6 million.
If all things were equal, Arsenal may very well have chosen to sign Michael, but they weren't. Michael was significantly more expensive than Källström and the Gunners didn't need to shell out a bunch of cash for an every day midfielder. They needed an occasional replacement, and Källström was that for just the cost of his wages.
Also, keep in mind that Bradley was already at a pretty big club in AS Roma. They were the second best team in Serie A and were going to the Champions League next season. He may not have been a starter there, but he wouldn't have been a starter at Arsenal either. It's not as if he was blackballed out of major European soccer.
There is no doubt that being an American doesn't help a player's ability to sign with clubs. The degree can be argued, but there's still a stigma attached to being an American that there isn't to being Swedish or Ivorian or Brazilian.
That said, there are stigmas to a lot of places. There's a stigma attacked to being Mexicans or Argentinean, as well as others. Those stigmas may not be quite the same as the American stigma, but they're there too.
That Michael is American may have played into Arsenal's overlooking him. We don't know, but the primary reason wasn't because he was American. There were a lot of reasons, but mainly, it was financial.