Closed Stable stopped working

You said there are six starbound.log files. If you take away the five numbered backup log files, that leaves the one starbound.log. I don't think Apple computers need to enable file extensions.

 

Delete the Starbound Folder, Verify it, then Reinstall it.

That should fix the problem.

 

Deleting the folder would trigger a redownload.
Verifying will trigger a redownload of anything missing or corrupt, which since you deleted it, would be everything.
Reinstalling would be again deleting everything and redownloading it.

You just listed 3 redundant things to do.