As your equipment is old, you may find any degreaser degrades the material used in the manufacture. That aside, If your assembly looks oily or dirty stick to your plan A, and the Halfords one is as good as any. However if it looks essentially clean, then read on.
I guess you want to maintain the originality of the machine, and not fit a modern switch. You presumably mean that when you press the start button and release it the machine stops, or tends to sometimes . It seems you can view the contacts involved as you are going to squirt something at them, so………If you examine the contact points in detail You may see that one has a spike on it, caused by arcing over many on/off actions. In that case, plan A will do nothing. There may be several contact pairs if the machine was ever 3 phase, so examine them all. Carefully remove any such spike, which may be quite hard. Don't remove more than you have to and leave a smooth/polished and clean contact. Don't bother about the hole in the other contact but clean it up if it looks dirty/sooty. Put all back together and try it again.
As already stated, ensure you are working on dead equipment whatever you do. If you do use a degreaser, then don't use more than is necessary and let any solvent evaporate totally before you reapply power. Hope that helps.
Dennis