Well you might be surprised, its possible that whatever you choose to polish the cabinet up with will help conceal the rings at the top. Mybe not make them go away, but at least make them less noticeable. And yes, that's what doilies are for.
But I bet it will look a lot better if it is polished and shines up. It won't be perfect, but then it does not need to be. It will be a nice original old radio. You could try spot refinishing if there are any real bad areas, but I would not think that would be necessary. Keep us posted as you go along.
Gilbert