My guess would be friction on the barrel arbor pivots or a weak mainspring. I know the symptoms you mention, I have an HMT that exhibited those and winding fully would get the balance moving lethargically briefly then stop. Then holding winding tension got it swinging and it eventually continued but the amplitude is low. That will get stripped, cleaned and checked at some point.
you know what you are right it should not make a difference, all i know is when the wheel stops, i take my finger and operate the wheel back and forth in tiny increments and i feel something rubbing on that spot, and i cant see what it is. i will have to test with another wheel or cock, i cant use the other one because that one is running perfect i have other movements that i can pull parts from. I buy as many waltham 1907s as i can.
AndyHull, I managed to get a lot closer look and found where the broken off extended pivot of the seconds wheel gear was still inside the post of the seconds brass disc. The post is riveted to the seconds disk. The pivot of the seconds wheel gear is press fitted into the post. It was a very tight fit requiring a tiny bit of oil, setting time, and needle nose pliers to remove.
Now, I have to find another seconds wheel gear to replace the one I broke.