Yes, writing tests work when you already identified the potential bug if future code changes break things. My first thought was you can't automate bad logic bug discovery.
But you almost can. There are tools like mutagen[0] that compile modified versions of your source mode, and if all your tests pass, then your tests don't actually test the logic that was changed. Now there's a lot of nuance to it (false positives, don't run this on stuff that can write files, etc.), but the basic concept is another good step towards automated bug detection.
There's also fuzzing, but that's (IIUC) not so much about bad logic but instead about finding novel (usually bad, memory driven) states.
Can you elaborate on this? I have never heard of any efforts to automate bug discovery beyond static code analysis tools in IDE's for example.
I agree with your first and last point though. Cheers