Why is it always "they have no baths?!"
Dude.
Everybody, every culture, every set of humans around the world knew the idea of heating up water.
There are baths made by cliff-dwelling cultures, by jungle-dwelling ones, by plain-, swamp-, island-, mountain-, and forest-dwelling cultures.
Everybody knows "bath," dude; and no lack of money has stopped humans from bathing; even ancient Ireland's people dug holes, filled them, tossed in hot rocks, and had a bath.
What is it with manga authors that they think bathing is unique? Europeans in the High Middle Ages bathed more often than most Americans today, and WAY more than the average Japanese.
Such a weird and dumb trope; inaccurate, too.