I commonly hear this, and yet it is so wrong. I don't understand how this idea that unit tests significantly protect against regressions has spread so far. Classic texts such as "Growing Object-oriented software with Tests" note, rightly, that end-to-end tests are for external reliability.

Unit tests, which don't operate on that abstract high level, give you relatively little safety. Sometimes they can ensure that an algorithm is not broken, but algorithms is usually not where the legacy code has issues. Often unit tests are at the level of "assert that X function calls Y stub Z times".