The potential for exploitation and coercion of weak players in the transactions is one reason. The same as with euthanasia.

Society feels there is a compelling interest.

Is that legit? That point is up for debate, since it's common for almost everyone to sell their bodies for money. We call it a "job". If that service you sell is done willingly and for acceptable compensation and you can manage your workplace security, it should be your choice, most of us agree. Roofing, for instance, is hard work. Easy to get into. Low barriers to entry. Looked down on by society, but agreed that it's essential. Not that safe. You can get workplace related disease and injury. Yet, we let people do it. A lot of people think prostitution should be similar. For the same reasons. And a lot of people don't.

Hell, for a long time, PINBALL was illegal. Society sometimes does crazy shit.

Historically, prostitution has been tolerated, not encouraged. Its relegation to illegality in the USA now is likely rooted in Puritanism, like so many other things.