A bot should do it. Because sometimes people throw a tantrum, so it's easier to just ignore a PR. Or a maintainer might post a quick reply, only to be bitten later on. I love and live by open source, but the drama can be exhausting.
"open source" doesn't mean "accepting drive-by contributions from unknown authors". "open source" means "open source". If you want to apply your patches, you're free to fork the repo.
Is there no way to just have a script add a comment on every new PR saying "Due to insufficient staff, your PR may not be reviewed for a considerable amount of time."?
Your code is important to us. Please stay on the line, and the next available reviewer will answer your code. Due to unusually high code volumes, this response may take longer than usual.
I think the issue is about letting people know that PRs won't be reviewed/merged before they bother putting a lot of effort into them (both the commit itself and setting up/documenting the PR).
This has always been a weird quirk of GitHub. You can disable issues, boards, wikis, but pull requests cannot be toggled. It's a pain point for lots of projects: those closed to contributions, those which are not primarily code (issues-only), those that use a different platform for review (e.g. Gerrit), ...
Why does GitHub persist? It's not like forcing the feature enabled helps anyone. The maintainers will still not merge if they don't want to. There are bots in the marketplace that will close PRs with a message. GitLab has a toggle. Why is this so important to GitHub?