I think that a bigger concern would be razor burn or irritation, which could possibly be aggravated by the strong chlorine in a public pool. I used to shave at the gym after I swam (before I stopped working out and regained 20 pounds. Sigh...). I used to get find a few guys staring at me with interest while I lathered up with real shave cream and a badger brush then shaved with a vintage DE.
My mistake on honeymoon last year... I was in the pool a few hours, I came home and decided to shave straight away. It was the most painful and brutal shave ever. Swimming pool is not good pre-shave prep.
I swim in a chlorine pool at least 3 times a week and shave with a straight almost immediately after. I use soap with tallow and have no shave or skin problems related to the chlorine (. . . or otherwise, fyi, since wet shaving).
Not too far from me here in the UK is the Droitwich Brine Baths, currently closed. The brine is natural in the area and was a big salt producing area. The town was always popular as a Spa with bathing in the natural brine, which is completely saturated and as strong as the Dead Sea!
Very relaxing to float in the warm waters. First time I went, I made the grave mistake of shaving that morning! As soon as my face hit the water, it was on fire! You don't realise how tender that skin is straight after a shave until you soak it in brine!