You're probably right about the source of the leak. The grease in the tonearm bearings is too thick to run out. You can replace the cueing device grease yourself but it is really a job a professional should do just to keep things clean and tidy. Or you can do without the grease and just manually lower the arm into place.

A new suspension belt is fairly easy to replace. Or you can upgrade the Rega and look at a DC motor replacement. The belt is a lot cheaper than a new motor.