It’s hard to find a postulated public toilet, when you really need to go. Sometimes they’re just not there.
The Romans (and their Romanized conquests) did pretty much everything in the public baths, including bathing (heh), exercise, scraping with the strigel, hitting the head, possibly even hair removal of different kinds, and, even though the public baths were kept gender-separated, having the enjoyment from prostitutes. It’s surprising they ever came out of there.