Every culture that eats beans has discarded their bean rinse water

Because there hasn't been a need to create an egg substitute that produces recipes that traditionally just used egg.

you don't really see recipes that have you cook the beans in their own rinse

I thought the soak rinse was always discarded, the aquafaba is from the cooking water (either in a can or the pot it's boiled in)

You can taste saponins, a quick google search suggests that there are definitely some people making extremely bitter aquafaba (i.e. likely with a high saponin quantity) but that's not supposed to be the outcome.