A swimming pool isn't a gigantic body of moving water that is already barely clean enough to be swimming in it in the first place.

I haven't been to a public pool in ages, but growing up in a small US town I can't say I remember any of the kids at the community center taking a shower before using the pool. It may be different now, but I would be disinclined to say a shower is universal at least based on my experience.

I thought running through the shower area between the changing room and the pool and trying to avoid the cold water as much as possible was the norm, but maybe it's different in France.

Yes, but pools aren’t natural bodies of water that already don’t meet sanitary standards, not to mention the water is constantly moving so it’s not like the water is worse when someone goes in later.