First of all, compared to later Singer machines the bobbin winding system on a Featherweight is not all that great. Several things have to be just right to make it work well. And the white machines with the black plastic bobbin wheel and motor pulley are the worst due to the reduced coefficient of friction (slipperiness) of the plastic.
The plastic wheel is press-fit to the winder shaft and sometimes is slightly cocked on the shaft, causing the OD of the wheel to wobble and not run true (this is called “runout”). You should be able to see this wobble if you turn the winder shaft slowly and watch the edge of the wheel. If you are careful you can reseat the bobbin wheel on the shaft and make it run more true (without runout).
If you still have the white belt then it is likely original. Several problems will develop with the old original belt. In addition to the belt becoming worn and slick, it may have also set in one position too long and "taken a set" around the motor pulley. This will cause a high spot in the belt that will push the bobbin winder away from the belt each time it comes around. That is why the winder jerks and does not run smoothly. This can be solved by replacing the belt.
Unfortunately you won't find a white belt as they have not been made in some time. When you buy a replacement be sure you buy from a knowledgable FW specialist that turns their inventory over often. There are good quality belts and there are bad ones. A while back all we could get were bad belts that were way too thick and heavy. That situation has improved in general (from the suppliers where most retailers get their belts) and a couple of other sources are now making better quality belts. The pinkish brown plastic toothed belts are actually better for bobbin winding as they are "stickier" on the OD and "grab" the winder wheel better than the black rubber belts. But many folks don't know how to adjust them and don't like the looks. Either will work.
When you look at a oiling diagram for the FW it shows to put a drop of oil on the pivot point for the bobbin winder. In my opinion this is unnecessary and a bad idea. You only need to lube things that move. A bobbin winder is supposed to spend its life either winding a bobbin, or the other 99 percent of the time just 1/8 to 1/4" off the belt. Oiling it so it will move that 1/8" makes no sense and only serves to eventually make the winder too loose to stay against the back of the belt. If yours won't get tight and stay put, take it off and clean the mating surfaces and washers of all oil and reassemble it. Don't forget to clean off the machine casting where the winder makes contact with it.
If you do all this and you still can't get a consistently tight and proper bobbin wind, then just place your finger on the winder where the oil hole is while you are winding a bobbin and hold it down against the belt. No you shouldn't have to do it but there are worse things in life.
Good luck!
David