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It sounds like you're basically getting "free height" from the air pockets, so it would take the same energy to raise a solid stream of water 10ft as raising a 50:50 air:water split 20ft, because they have the same weight. Am I understanding that right?
Yeah, absolutely. It's not the only thing that's going on but that also happens.
So for example how they'd combine is, you could have 10 coils with a one-foot column of water in each, and that gives you 10 feet of pressure at the output pipe. That's way more than you would expect from the whole system being just one foot high (or a little more in practice) and self-powered. But then also, because it pumps a mixture of water and air into the pipe, it can reach a destination 20 feet up, by using that 10 feet worth of pressure (because as you say it's only lifting 10 feet worth of water and the weight of the 10 other feet of air is negligible).