There is this story, that on the 1936 Olympics in Berlin, the finish athletic team arrived and started to build a sauna, because it was missing.

As a Finn I can agree with this sentiment. It's a cultural thing. And we are not a filthy rich nation.

Really? I doubt that. Perhaps most buildings, as storey buildings do often have a shared sauna. As an anecdote, I have lived perhaps in 6 rented apartments, and two of them had a sauna, and I don't think I decidedly chose non-sauna option.

Here's some article in Finnish about the current trend that new apartments don't necessarily have a sauna: https://yle.fi/uutiset/3-9393813

It could be that most households have but also I've seen an article that people use them plainly as storage rooms.