That seems like it must be an exaggeration. Not once in my entire life have I seen a beach closed due to any of these factors. I know it happens, but I don't think it's as common as you make it sound. The only time I've ever seen a beach closed was due to rough surf.
Of course, I don't claim to have been to most of the beaches out there, but I've been to a few! Have I just been getting really lucky?
I've been going to the beach several times a week for about 13 years, even during winter. Never seen it closed. They monitor water quality regularly and the results are readily available.
You're lucky. I live near the Baltic Sea, a relatively shallow sea with a very narrow link to the ocean, and agricultural runoff and sewage from 100 million people drain into it. We have very regular algae blooms, typically late summer when the water warms up sufficiently (for some reason, even though it was very hot, very little this summer).
The sewage problem is largely solved nowadays, thanks to modern sewage treatment plants, but thanks to the disastrous EU agricultural policy (CAP) there has been no improvement at all in the agricultural runoff for over 20 years.
Sure there are always exceptions, and it may have been just some unfortunate phrasing, all I know is that the phrasing explicitly didn't specify that the river being unswimmable was at all an exception.
It's like if you asked a lifeguard if it was safe to swim and they replied "Many people survive".
Of course, I don't claim to have been to most of the beaches out there, but I've been to a few! Have I just been getting really lucky?