Why did public baths go out of fashion?

They went out of fashion because the infrastructure to run them -- the mechanisms that brought them water, that heated their water, that separated out the different heats of the various pools -- required an enormously sophisticated and complicated infrastructure, which the Roman Empire had. But when the empire started to fall apart, people couldn't maintain that, and the invading barbarians disabled the aqueducts. There was never an empire large enough to support that again.

That is mostly wrong. Public baths remained popular in the big cities until the church worked to suppress them as centers of immoral behavior. It probably did fade out in the smaller cities because of the lack of infrastructure. But the Catholic church actively promoted not bathing on a regular basis for moral reasons and that has more to do with why even bathing at home among the rich was not common.