Banyas can be anything from luxury to very old and basic, right?
Right.

One thing I'd love to do is to sit in the sauna for a while, and then take a quick dip in an ice opening, in the Winter.
Must be fantastic for the body, circulation and all that.
This sounds awesome. Unfortunally banyas are located within private gardens or at a particular site of a town or city so they're rarely anywhere near creeks, rivers, etc. So the most you can do near a banya in winter is take snow baths

Personally I like banya better than sauna, even though saunas are usually equipped with swimming pools, entertainment and stuff.
Sauna feels plain and soulless, maybe, I dunno how to describe it.