I doubt Cap would have moved to DC-Earth because the people would have "worshipped" him there. It would be more of a case of which world needs him more. Frex, in MC2 Cap stayed on an alternate Nazi-ruled Earth after destroying that Earth's universe-hopping tech (to protect MC2 Earth) because freeing that world was the right thing to do.
I think it's funny that when the Avengers see how loved the DC heroes are (the Flash Museum, Superman getting an award from the UN, some girls winning a Supergirl fan club contest), they immediately conclude the Leaguers have taken over the world and make the people worship them.
And the Flash landing in the middle of a torches-and-pitchforks mob trying to murder a mutant kid probably wasn't the greatest introduction to Marvel-Earth.