It is definitely that they were allowing Americans to gamble.
"would a gambling site that didn't take a rake and just ran on ad revenue be allowed to have American customers?"
This definitely would not be allowed.
You mentioned the money laundering charges a couple of times. Those were ancillary charges. The primary charge was that they were operating an illegal online gambling business i.e. letting Americans gamble on the internet.
"I'm an American citizen. Would I definitely be allowed to use Betable?"
It is not illegal for an American company to offer real money online gambling to citizens of other countries where it is legal and regulated. So Zynga could get a UK gambling license and offer online gambling to UK citizens. It is a long, slow, difficult and expensive process - Betable has already done it and is allowing others to leverage their infrastructure (legal and technical).