There is an industry based on providing short term lodging for people that is subject to oversight and regulation. Businesses operating in this industry are usually called "hotels".
http://www.nbcnews.com/id/25243594/ns/travel-travel_tips/t/h...
Your point stands though; businesses have the right to turn away customers for any reason that isn't discriminatory or class protected.
Perhaps there's a reason to consider regulating certain platforms as utilities once they reach a certain scale? (Not referring to AirBnB in this case)
People are outraged Airbnb can ban you without recourse -- tech must be regulated! Not a peep against hotel giants who can do the same thing
People are outraged that Amazon wants to build an HQ2 in NYC because it'll displace people and must be stopped! Not a peep that it's home to the headquarters of 52 Fortune 500 companies already and grows by tens of thousands of people a year.
People are outraged that Apple paid little in income tax on a huge profit last year and tech must be regulated! Not a peep about Chevron, Delta, or Halliburton.
It's just populism. People know and interact brands like Google, Apple, Airbnb, etc so they're easy targets, and as a result we're going to end up with regulation that narrowly targets that high profile offenders without doing anything to solve the broader root cause issues. California's AB5 is a perfect example -- it's basically meant to target Lyft/Uber/etc and exempts many industries for no reason: hair salons, insurance brokers, real estate agencies, photographers, fishermen, and more. Why? Because they want to specifically target the big bad tech companies without fundamentally fixing the problem of ensuring all workers get a living wage.
Sure, I suppose nothing legally stops them, I guess, but it just doesn't seem to happen. (Perhaps there is a legal barrier, where they try to leave a defensible paper trail so as to show it wasn't e.g. racially motivated, but then, I don't know why such concerns don't spur similar behavior from Airbnb.)