Funny, I was going to mention the Packers. They are the exception to the rule. A large reason for their national recognition is due to their history. They have a large local draw of course, they're the only show in town. Across the nation they have a following due to a deep and rich history.
As for the Patriots and Chargers, are San Diego and Boston global cities? Yes they are. Although the Chargers aren't really a national team. Same with the Browns, they're not a national brand. Patriots also have a large bandwagon following, but Boston is most certainly a global city. While I know they don't physically play in Boston, its essentially Boston's team. They also included the entire region. New Orleans is on the cuff as well.
Even still, that doesn't mean the Rams can't become a national brand in St Louis, its just easier and quicker to do in LA. Of course winning speeds things up, but so does playing in a large market. Also LA fans show up just fine, I mean crap the Lakers were the fourth worst team in the NBA last year, AND there was another LA team that was the 4th best, and they still averaged 98% attendance.