One thing that confused me. Does not being THE Alpha timeline make you doomed, or are you only doomed by not closing all the stable timeloops? Could a timeline, in theory, un-doom itself by closing the loops?

You can’t fix an unstable timeloop without creating a doomed timeline. Because in order to stop the timeloop from becoming unstable you’d have to go back to stop yourself from doing what you did which would in of itself create another timeline. One where you were doomed and one where your doomed self stopped you from becoming doomed.