The swallows I have seen can eat a fairly big insect for their size, but I think a "water bug" cockroach might be too big. A purple martin probably could, though. Cockroaches are quite nutritious, so if it were flying in the daytime, it would probably be popular with these birds.

Swallows generally feed on flying insects and catch them mid-air. It's unlikely that cockroaches are found within the areas that they hunt

I'd be astonished if one even tried. They're simply too big when they can fly.

I mean American cockroaches. The huge flying ones that are also mistakenly called "water bugs" and "palmetto bugs" in Florida. I hope a bird specialist or an entomologist can answer accurately!