I don't put chlorine in my bath at home, but it would absolutely be required in a public bath. Other health issues aside, I'm not sure this would be a net positive for the environment.
There's no need. In Japan, one must fully wash oneself before entering the bath; if you had diarrhea, say, or some other pressing need to void, it would come out before you got into the bath itself.
In what way would this be an environmental win, then? I went to one onsen and between the showering before and after, I’m pretty sure I used more water and energy than showering back at the hotel.