Fair enough, however the Fermi Paradox and the consequently postulated Great Filter are a thing precisely because the evidence of this competition should be present but is not.

> The local group is bigger than the galaxy.

You are certainly correct, I misremembered the terms. Substitute for the local regions of our galaxy: https://en.wikipedia.org/wiki/Local_Bubble https://en.wikipedia.org/wiki/Orion-Cygnus_Arm