There's a couple of thoughts to share regarding the issues and possible repair of the damaged thread.
1. This problem happens quite often on small diameter threads and fine pitch threads. A common practice to develop and I use it religiously is when starting a small diameter fine pitch screw, first set the screw in the female thread and place the driver on the screw. Turn the screw counterclockwise and feel the start thread on the screw drop in to the start thread of the female thread. It will make a slight clunk. Then rotate the screw clockwise and install the screw. This prevents the screw from trying to start cutting a new thread in the existing female thread. Try it. Once you understand what to feel and engage the thread you'll buy into the practice. This is especially true when starting self tapping metal screws in to plastic female threads which had a screw in them before. How many times have you removed the screws from a plastic housing and then on reinstalling had some of the screws go in nicely and then a few went in really tight? If so you know exactly what I mean by cutting a second thread in an existing hole.
2. If by chance you encounter a bad thread and want to chase it out with a tap, do not use a starter tap which was designed for tapping out drilled holes. Those taps have a long lead in taper and if you use one, it will be really hard to get the tap started in a damaged threaded hole. The right tap for the job is called a bottoming tap. It is really designed for tapping out the last thread in a blind hole. If you think of the starter tap with that long lead in taper on the starter threads, if you tapped a blind hole with the starter tap and then ran a bolt in the thread, the last couple of threads on the bolt would be squeezed in the bottom threads, so tapping a blind hole really needs the starter tap to tap as far as possible then finish the thread with the bottoming tap. What's really good about the bottoming tap is the lead in thread is only the first thread and a half on the tap. When you try to engage a damaged thread usually the primary thread or the original thread is deeper than the damaged portion. The bottoming tap will feel out this original thread and start cleaning it out as you engage the tap in the damaged thread.
Hope this is useful info to some of you out there.