Earlier quoted context omitted.
How do you know it's working if it doesn't have unit tests that exercise edge cases? Or even basic functionality? Also, good unit testing can allow you to reduce the time it takes to write software by giving you an isolated environment to run your code. Unit tests are a tool to get you to working code.
No. Failed tests can only tell you that you've found a problem; passed tests can't tell you there aren't any problems present. Provably correct code can be proven correct.
Using tests as a way of finding problems seems like not a great way of saying code has issues x,y, and z but they are a great of saying something does a, b, c. If you are going through a refactor you might not be implementing a, b or c correctly anymore and if your tests are designed well this will be caught.
If we are only talking about bug catching, you are right there are better tools. But I'll say if somebody says code does something and there are no tests demonstrating that fact, I don't trust it generally unless it's something like a game where it's hard to test certain things.