Do berghia actually get rid of aiptasia?
In our experience, yes. Berghia eat aiptasia and nothing else, so they keep hunting until it's gone, including the ones hiding in the rockwork. Four berghia cleared about six aiptasia from our 16 gallon in about a week, and we haven't seen one in that tank since.
What happened in our tank
About six months ago, our 16 gallon had about six aiptasia. We added four berghia, two large and two medium. We watched them the first night and then never saw them again (which is normal: I added berghia and now I can't find them). One by one, the aiptasia disappeared. The whole thing took about a week, and none have come back.
Watch one eat
I was sure this aiptasia was way too big for one little berghia to take on by itself. It was about four times its size. It went for it anyway, and you can watch it fill its stomach and see the brown from the aiptasia flow out into its cerata (the feathery parts on its back).
One small berghia taking on an aiptasia about four times its size
Why they work when other things don't
Berghia only eat aiptasia. They don't bother coral, fish or your cleanup crew, and they can't get distracted by the food you feed everyone else, because they don't eat it. They're also small enough to get into the rockwork, where the aiptasia you can't see are hiding.
What they won't do
They won't work overnight, and you probably won't see them doing it. And once the aiptasia are gone, berghia have nothing left to eat, so they aren't a forever pet. If you're comparing options, here's how to get rid of aiptasia (what we tried, what we skipped, and what worked).