A private business that serves the general public is a public establishment though, even if it's run out of your home. The criteria is whether you restrict access to some private group, not whether it is a privately owned space. I'm not seeing how a server that responds to any traffic without any selectivity (KYC, basically) is not analogous.

It doesn't matter if you're not required to go to the frat house. It matters whether the frat house lets the public in while exhibiting their fornication, or has filters at the door.