The best suggestion I've heard recently is, when someone writes a flaky test, that person needs to be the one who fixes it. (If you write flaky tests and I fix them, I learn how to not write flaky tests, and you keep on writing them, blissfully unaware of the pain that they cause every day.)

If only one person is writing tests, that's a problem you won't have, but what's worse... I think you have it worse.