> Notably, they were only intended for use by men.
this complicates public bathrooms quite a bit, and ADA requirements do too... once you've got a place to sit and doors on rooms you've got the vagrancy and drug use problem
though China does ok with the model of a big room with holes in the ground and almost no dividers or affordances at all (everyone brings their own toilet paper, old people bring their own chairs)... they have public workers to hose them out
Sounds like the Romans had it better than your description of the Chinese, though I am sure in the modern world someone would be stealing the sponges for unfathomable reasons.