In Budapest I spent an afternoon at a great public bath with many inside and outside therapeutic pools. Is there anything like that in Rome, Florence or Venice?
Venice--no for sure, unless it's a super posh hotel's spa.
There are some closer to Rome:
https://www.romeing.it/natural-thermal-baths-near-rome/
However, it looks like these are of a smaller scale.
Here's another webiste: https://www.nationalgeographic.com/travel/article/paid-content-italy-most-iconic-thermal-baths.