Linkcheck failures are an annoying to many PRs.
Most linkcheck failures are spurious and transient, and even those which aren't are generally not caused by changes in the PRs which they block.

So I would like to make them a periodic check instead of a PR check.

This leaves the question of how bad it is if you do introduce a new, broken link in a PR -- which is then not picked up until later.
I'd suggest that just isn't much of a problem