Why unit tests?

In conversations around unit testing, I regularly hear someone assert the real reason we write unit tests. Sometimes the conversation goes like this:

Alice: "We should write unit tests"
Bob: "No. We can already achieve result X through some other approach." or "No. Unit tests fail to accomplish Y which is important."
Alice: "But Z is the real reason we write unit tests."

It would help me to have a list of supposed reasons for unit tests, so I'll catalog them here.

Note that I'm not asserting specific definitions of "unit" and "test" and "unit test", although varying definitions here are a big part of the problem.

Correctness: proof that my code does what I intended it to do.

Detractors: There are many bugs that unit tests won't catch, e.g. integration bugs, security bugs, timing bugs. Also, the mindset that created a bug is also the mindset creates the test, so we can't rely on unit tests to catch all the bugs.

Proponents: Developers often fail at basic correctness, leaving testers the tedious work of finding easy bugs. By using unit tests to ensure basic correctness, even imperfectly, testers can do the interesting and important work we need them to do.

Regression: proof that my changes didn't break something else.

Proponents: Much time and energy go to fixing regressions, which could be eliminated if we had tests.

Detractors: Same arguments as in correctness.

Refactoring: I can safely refactor without introducing regressions.

Proponents: Refactoring is key to the long-term well-being of the code base, and the mental health of the programmers. Having unit tests in place makes that safe.

Detractors: The obvious arguments from above, plus the burden of dealing with failing tests whenever you change something that should be innocuous. Do you fix 100 tests or just throw them away?

Design: unit tests help me reduce coupling & increase cohesion

Detractors: I don't see that happening.

Proponents: When a test is hard to write, that's design feedback to refactor your code. When the test is short, clear, easy to write, easy to read, has a good name, and runs fast, you know your code is in good shape.

Note that it can take a lot of practice to develop the skills required here.

Efficiency: Unit tests affect the time it takes to get the job done

Detractors: Time spent writing unit tests could be time spent creating customer value. Also, writing these unit tests takes a long time.

Proponents: Highly skilled unit testers write less code because they have less redundancy and they only implement as much functionality as is required by the tests. They also spend less time fixing bugs, and can afford to perform root cause analysis on every bug.