I'm going with the points. You said you're missing s screw on the points plate, and you've "polished" the points with emery cloth. You never said you used a dwell meter.
The points gap is just a ballpark, the real tune-up parameter you need to pay attention to is the dwell. Dwell measures the time the points are closed, and I have many times set my gap to the gnat's ass and then checked dwell to find out the dwell is WAYYY off. It makes a huge difference on how it runs if they are way off. Adjust a NEW set of points using the gap to adjust the dwell to within the specified range. Larger gap=lower dwell angle, and vice versa. When the dwell is correct, forget what the gap measurement is.
Also, I've read somewhere that you should'nt file points, don't ask me why. But it's worth the effort to figure out HOW to get a new set of points in there, even if you have to scab in just the new contact arms. You really need to eliminate the points and point gap (dwell) as a problem before you move on to more obscure (and expensive) fixes.