The general idea with no swimwear in the sauna, is that it soaks up chlorine from the pool and your sweat when you sit in the sauna.
So aside from bringing in a chlorine smelling swim short into a sauna, you will also take out a lot of your sweat that soaked in your shorts into the pool.

It's unhygienic really.

I'm Dutch, nobody bats an eye on a co-ed sauna/spa here. It's actually a very relaxing day out with friends as a lot of these places also serve very good wine, beer and dinner. Which you can enjoy by just wearing a bath robe).

There's a 'swimwear' or ladies only day at a lot of them once every other week or so, but it's not as popular as just going co-ed in the buff. Plus kids generally aren't allowed in the evenings if you want to enjoy some peace and quiet.