I think pleasecalllater's point is that without unit tests, you'll spend a lot of time just pinpointing the bug, since it could be anywhere in your module (which unit tests would cover), not just at the interface to other modules (which integration tests cover).
- run fast (<10s)
- run on your local machine
- run on your local code (including uncomitted diffs)
... whether they're called "integration" or "unit" tests doesn't matter, as they will detect regressions at the earliest possible time.
Unit tests might guarantee smaller code search area (because maybe the failing test only executed 1% of the code), but if running "git stash" makes all the tests pass again, I have a pretty good confidence about where the error lies.