You're mostly spot on but your last sentence is flawed, as it's chloramines that are the problem with pools, not chlorine, and urine is all-too-often the root cause
https://www.chemicalsafetyfacts.org/health-and-safety/how-ch...
https://www.chemicalsafetyfacts.org/health-and-safety/how-ch...
> perspiration, oils and urine that enter pools from swimmers’ bodies.
I choose to be an optimist and will believe that list starts with most common source and ends with the least common.
It seems like that would still be happening in the absence of urine?