I would think quite a few birds would eat them, whether domestic or wild. amphibians and reptiles, too. basically, all the things that would also eat earthworms. there are proven strategies for encouraging populations of all those potential predators.
I've sort of leaned toward encouraging rove beetles in the past on the assumption that more diversity is generally better. and I still lean that way. my guess is that over time, the populations would balance out as rove beetle predators get the picture and move in, particularly if somebody encourages them with inviting habitat.
some friends mentioned that they had a terrible slug problem for a few years, but plenty of earthworms. then snakes moved in and ate the slugs and earthworms. then herons moved in and ate the snakes so the slugs and earthworms rebounded, but not to their previous high levels since a few snakes evaded the herons.
diatomaceous earth would probably knock out the beetles, but it would also knock out a great many other critters that you probably want around, possibly including the worms.
and what the presenter said might be true, as well: a huge number of worms isn't all good. some decomposing vegetation that remains on the dirt for a while has several benefits. too many worms will remove that cover too quickly. I believe that's the main complaint about earthworms expanding their territory: destroying the duff layer in forests quickly decreases biodiversity, at least in the short term. a reasonable population will still incorporate vegetation in a garden into the dirt effectively, just not quite so quickly. so maybe the early worm boom is
enough to accomplish an initial dirt improvement before the rove beetles show up, then the lower population will continue to improve things along with the greater diversity of critters that show up.
and keep in mind that rove beetles eat up young slugs and plenty of other unpleasant garden critters, too. they aren't just eating worms. they're also shitting all that out and cycling nutrients.
run some chooks or ducks through periodically. encourage some snakes and frogs and birds to hang around. earthworm populations might reliably take a dive once rove beetles show up, but they won't disappear entirely.