My clients usually get in before pushing. Whenever they want, as long as they're in active labor.
In my experience most people do not have enough hot water to fill the pool that comes with the heater, so you end up having to heat water on the stove regardless. So it actually ends up being worse with the big pool with the heater, because you find yourself rushing back and forth with four pots of boiling water just to fill the pool up in the first place rather than a pot or two here or there to maintain water temperature. The smaller, oval pools like La Bassine or Birth Pool in a Box also have less surface area to lose heat through but still plenty big enough (and the size and shape make more sense to me anyway; after all, your body is oblong, not round, and the big round pool is just a waste of space and water and makes it harder for someone to assist you without). You can always throw a tarp over the top to keep heat in if you get out.
They had to drain the pool first in order to make the water hot again? That seems excessive. You should be able to keep the pool warm by adding a little hot water now and then. I really don't see the big deal in that, anyone can do it and it's not a hardship. It's not like anyone's asking the mom to get out and do it herself. It's done quickly and quietly without disturbing the mother, if it's even necessary at all.
Then again, I don't think it's necessary or even in the mom's or baby's best interest to maintain the water at a high temperature.