The fact that your tests are flaky at all means that the integration tests have revealed a bug that you need to fix. The tests have done their job. The fact that people aren't looking at the results speaks to a larger cultural issue.
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.