> At the same time, we'll eliminate drive-by pull requests that also need reviewing.
I don't think these are good reasons to move away from GitHub. If your project is truly popular, then the people who are filing these issues will follow you to your new home. You'll get a short period of respite, and then you'll be back where you were before. If your project is only popular because you're on GitHub... then moving off will not only give you a respite, it will also destroy your user base.
This is why full-time software engineering teams have PMs. They act as an API for the engineers: helping others understand what sorts of requests can be made, what kinds of responses can be expected, and turning away abusive/malformed traffic.
It simply sounds like this project is popular enough (and >5000 issues in just over 2 years certainly qualifies as popular in my book) that it needs non-engineering attention. Get someone to join the team who wants to fill in a PM role.