I prefer keeping the labeling as is, so core devs PR get "awaiting merge" label upon creation.
If a core dev actually want someone else to review it, I suggest using the "request review" feature, and select another core dev to review the PR.
Do the people wanting other core devs to review typically have someone they want to review, or just anyone to do a review?
If I'm specifically waiting for feedback on a PR, I'll generally put an explicit "WIP: " (for work-in-progress) in the title to warn other folks not to merge it yet (it's a habit picked up when using GitLab, as that will actually prevent merging if you do that).
We could set up a separate bot at this point, but with the new automerge bot this won't be such an issue as people will simply not add the "automerge" label until they are ready.
So if folks don't want to make "No label" the default, then the convention could instead be to add "WIP: " to tell Bedevere to skip adding the "Awaiting merge" label.
I think we may want to hold off on any changes until we see what happens to our standard practices with the automerge bot.
-Brett