I'm not disputing that people lived in cold climates. That is simply not the same as jumping in an icy pool by choice, with the expectation of some health benefit.
Last I checked, cold wasn't for pain management, often the cold exposure hurts worse! If anything, it is to stop swelling - but swelling is a sign of the body immobilizing the area in an effort to protect it during healing, while it is bringing all of its other healing mechanisms to bear on the area. Applying cold would slow that while process down, actually retarding your own body's healing efforts.
You mention bathing - how much bathing was actually happening in cold waters? Hot baths in sweat lodges were more common around here (pacific northwest). Not sure how much bathing in water the Inuit do, probably not a lot. But this is where I am making guesses, and don't know the answers. For all I know the Inuit yelled "cannonball" and did polar bear challenge plunges all the time!
Roman baths were also heated, as I understand it. Very different climate there.
Everything I have come to understand about survival would point to avoiding 1) cold and 2) being in water whenever possible, just from a general sense of risk/reward.
I still don't hear any evidence of ancestral humans or any animal species practicing cold plunges for physio- or psycho- wellness.
