[eesmith]

A "public bath" isn't the same as a "public place", which has a wide range of meanings. Think of it as "open to the public", similar to how a pub="public house" even though it's privately owned and people can be kicked out.

A public bath has an entrance fee and staff, including cleaning staff.

While a train station doesn't have an entrance fee, even to access the elevator.

The streets are even more "public", in that speech like preaching has very few restraints. But a public bath has no obligation to allow yelling and preaching.

One of the ways people do public baths in the US is to join a gym. Another option is to go to a spa. Those two provide most of the market need for a public bath.

Other cultures also integrate a social experience in the public bath, but that is coupled to historical reasons not related to your objection.