This was, in fact, one of the actual mechanisms brought to bear against Backpage. Certainly the indictment was motivated by, you know, underage sex trafficking, but the enforcement mechanism used to achieve a verdict was mostly money laundering. You can read the indictment yourself: https://www.justice.gov/file/1050276/download. How many counts of trafficking vs money laundering do you see?
And finally, no, they are not "selling different content through other processors." These processors' entire business model is to create "legitimate," totally unrelated transactions to the credit card authorities, to take a cut, and pass back the funds to the merchant. I believe Backpage was instructing people trying to purchase ad inventory to go to a portal to purchase dog food for a while.