Iceland runs plenty of these and has a nice culture of frequenting the public bathhouse. It’s one of the few things you can do that is actually affordable there.

They do have the advantage of having essentially infinite clean energy in the form of geothermal heat. As do Japan in many cases, for that matter. I’m sure that has something to do with these institutions having staying power there.

Anyway, I think this idea has merits, but not as an energy saving measure. The reason for this is that in order to maintain good water quality, you have to shower thoroughly before getting into the bath, negating the potential energy benefits of the initiative. We can bring it back for it being nice, though!