I'm not saying your solution wouldn't work, I'm saying that any implementation of your solution would be an unreasonable burden. AirBnB is not a real estate agent and does not act as one, there are no laws against them listing properties nor should their be.
I don't see how you can say that and not feel the same way about payment providers. If AirBnB is not on the hook for activity that happens explicitly and exclusively on their platform I don't see why we should hold banks accountable for illegal activity that happens on their platforms.
Banks shouldn't be held accountable for illegal activity that occurs on their platform unless they are complicit in the activity neither should AirBnB. If the government comes to AirBnB or a Bank with evidence of criminal activity they should then have act but neither should be in the business of policing on behalf of the government.
I would agree with AirBnB having to be audited so that the gov't could do the policing but that's about it.