It sounds like you have a bad winding on the starter armature. If the starter stops on this spot, it will catch after repeated 'bumps' on the starter switch - or maybe not catch at all!
The starter switch issue IS a coincidence however. If you are certain you are bypassing the starter switch correctly, I'd pull the started and get it checked at a shop. You can also just cut to the chase and get a rebuilt rebuilt starter.
Unless a connecting wire is obviously broken, you can't diagnose a armature winding issue easily at home.
