Yeah if they got not food they should vanish with in 8 weeks or so.... I would say any stick is considered a food source besides maybe birdsnest and digitata...
dont think that you can just let them kill the rest of the SPS and then they will starve. You have to manually remove all SPS from the system (and kill what you cant remove) for the AEFW to starve (which takes a couple months). If you leave even a bit of SPS, the population will just reduce and they will resurge when you reintroduce SPS. They can live in a kind of equilibrium with coral, keeping it from looking good or growing well but not killing it (survival mechanism).
It will definitely take them a long time to vanish, and realistically it's not something you want to chance. You can either wait a long time and hope to have them all dead, or you can manually remove them.
Wrasses eating them is like peppermint shrimp eating aptasia... yes they can but they won't always.
Any fish eats them once there in the water column.... There is no predator or naked eye that can spot them on a coral that's the problem!!!! They blend very well a wrasse might catch one here or there but if you dip a infested colony hundreds if not thousands come off it and fish don't see them since there attached to the stalks and very flat.... Most fish can't even get into the colony to even try and eradicate them.... You gotta remove and aggressively dip everything for weeks while the tank is fallow.....