I have created a bot that I hope we can use to host all non-CLA checks for a PR that isn't covered by CI: https://github.com/brettcannon/bedevere. To start, the bot would check for a bugs.python.org issue number and set a failing status if one isn't found, else link the status to the issue itself.
If the approach I took in the bot seems reasonable to people I will add support for a "trivial" label which would cause the bot to say the check is successful due to the fact that the PR is trivial enough to not warrant an issue to begin with. We can then continue to expand this bot to do other things like check that there is a NEWS file (unless marked as "trivial").
I don't want to role the CLA bot into bedevere since I created the bot in such a way as to be useful to other organizations on GitHub. So the Knights Who Say Ni will stay a separate repo.