Your sentiment is strange to me - I've only ever seen unit tests as absolutely worthless. The reason integration tests are flaky is exactly the same reason they are valuable.
If I had a dollar for every time I asked someone why they weren't fixing their shit, and they answered "Oh, I thought the build was failing randomly again." I'd have a lot of dollars.
Yes, there are bugs that look like bad luck, but are really flaws in your logic. I have fixed a lot of bugs that look Random, but eventually you're left with, for instance, selenium just refusing to click a button.