Regarding oil-filled electric radiators

[Fumesucker]

1. Probably one of the heating coils has burned out..

Unless you see evidence of a leak that's the most likely diagnosis, they usually have two or three heating coils inside, low is one coil, medium is the other and high is both at the same time.

All the oil does is transfer the heat around, it's not consumed any way.

The other possibility is that the switch has become damaged and just isn't making the proper connection any more, I would take it apart and try to see if I could fix it but if you aren't electrically knowledgeable that's not a course I would advise.