American living in Copenhagen here. We check all of these boxes. The entire channel that runs through town is swimmable thanks to decades of work cleaning it up. On a warm summer day, you could expect maybe 1000+ swimmers in the water in the city centre across all the locations. We also do Vinterbadere (Winter Bathing) throughout the year. The only time you can't swim is after a day or two of sustained rain and there…
Why you can't swim after the rain? Does the water get dirty, or else?
1) Naturally, rains wash out street grime and small pieces of rubbish which may easily bypass the sewage system altogether and end up in the river.
2) Cities have systems in place to treat wastewaters before releasing them into the rivers. However, these systems need to store yet-untreated water in reservoirs, and heavy rain may fill these up. Then, the only solution (if you don't want sewage to blast out sewer holes) is to release the water without treating it. In fact, one of the main upgrades made by Paris for the recent Olympics was a massive, 30-meter-deep new basin [1] built exactly to mitigate that risk.
[1]: https://www.reuters.com/world/europe/giant-reservoir-ready-s...