There's no perfect solution and all of this still requires a fair amount of vigilance to get it right. But I would still argue that if you are proposing a PR, you've done the work to verify your changes work, tests pass, etc. If you later need to rebase your PR, you still should run tests, smoke test, etc.

Small issues can slip through, but on teams I've worked on, submitting a PR that is truly broken is really bad form. Sometimes that does mean submitting PRs is tedious, but that's just how it goes sometimes. Thankfully, it's not that often in my experience.