It's not a mess if you pay attention to where you actually give birth. Me personally, I've not ever used a pool, but my understanding is they are relatively easy to drain with a hosepipe and submersible pump and then you just scoop out the liner (which would contain anything that didn't drain).
The placenta is handled same way as if you're in hospital. If you want to keep it for whatever reason, it's yours. Otherwise it's bagged up as bio waste and taken away by midwives, same as any other waste, like gauze, gloves, etc.
I've had two home births, first one I ended up actually giving birth in a totally different room on a different floor than planned, so everything to actually cover the floor was downstairs and I was upstairs, but at the time, no one thought to go get all the stuff and put it under me. So yes, was a bit messy, but a carpet cleaner (cream carpets!) and some Vanish and dh's persistence and you'd never know a baby was born in there (unless you took some luminol to it and then would probably look like a murder scene! 😂).
Second home birth, I stayed where I planned to birth, put a cheap argos shower liner over the floor and an old duvet and some blankets - think about what you'd do if you were having puppies - and it was fine. No mess, the midwives must have just rolled the whole bundle up, and I never saw it again. They took it away with everything else. 15 minutes later and you never would have known a baby had just been born there.
If you think of birthing in hospital, those rooms get turned around quite quickly for the next woman, so they scoop everything up just the same, but have to do extra sterilising because it's not a home environment. The hospital pools are actually more of a pain to clean as don't usually have a liner.