That's an impressively burned antenna! Yes, grease build-up on the waveguide window and also running empty / with dry contents can cause that. All is not lost though. They are replaceable, a simple pull-off, push on. You can find them on ebay (search 'magnetron cap'), pick one with the same shape hole (triangle) in the top - I don't know if this is strictly necessary but there must be a reason why they come with different hole shapes.
It sounds as if the magnetron itself still has plenty of output power available!