What material is the white pip you have indicated, it should be an insulator, original ones are something like Bakelite. the arm should be a spring fit on it as well.
The condensor is inside the armature, and not something you can readily change, symptoms of this failing are loss of spark when hot, and by the sound of it you do not have this problem.
Chris.
If you follow the design of the points assembly, the rocking points arm is connected to the springs, the other end of the springs is connected to a brass pillar which is part of the brass backplate, the brass backplate has a second pillar which carries the pivot end of the locking plate, The other end of the locking plate presses down on the 'pip' which is a push fit in the rocking arm. That's a complete circle with all the metal parts at the same potential (earth). So the pivot does not need to be an insulator from an electrical function point of view. However, as just mentioned the pip is a tight fit in the rocking arm so, as the arm rocks, the pip grinds backwards and forwards against the underside of the locking plate every time the points open. That calls for different materials so yes, originals were a plastic of some type - not sure what.
They are often missing and we make them from PTFE rod. Not all rocking arms have the same diameter recess to take the pip so the first job is to turn the PTFE rod to the diameter required to get a press fit in the rocking arm. Then just part it off. As the parting off blade nears the centre of the rod, the new pip tends to deflect away from the blade. Keep going and it will eventually part off but with the advantage that that deflection results in a raised pip in the centre which is absolutely ideal to fit into the groove in the underside of the locking plate!
Starts and runs when cold but difficult to start when hot is a common problem with a magneto. Leave it to cool down and it starts and runs OK again.. This is an early warning that insulation is failing somewhere. Ignore the warning long enough and the problem will get worse and eventually, even if it starts OK, it will let you down further down the road. Insulation in a magneto deteriorates with higher temperatures and with age. It could be condenser, coil, slip ring, pickup(s) HT leads, plug caps or any combination of them! Yes, original Lucas condensers are now long past their best and, to be honest, we don't even bother to test them - they are automatically replaced. We always rewind the coil as well. It may look good on the outside but when cut open, original windings are always very green with verdigris. Testing slip rings and pickups while cold is quick and easy - if they fail then they are replaced. If they appear to be OK, we put them in the oven and test again while hot. It's not unusual to find that these Lucas parts fail when cold and much more often when hot - they are usually replaced as well. The fact of the matter is, all these parts, if still originals, are 60-70 years old. Materials and manufacturing technologies have vastly improved in that time so, once replaced with modern alternatives, the magneto should see most of us out! I also mentioned HT leads and plug caps but, to be honest they are far more likely to have been replaced already and rarely show problems.