I agree that the Ground frequency becomes overloaded at times and it's hard to get a word in edgewise.
Having a 2nd Ground controller/frequency is an option, and I'm sure it will be brought in by the time the parallel runway is built.
We have moved towing requests initial radio calls to the clearance delivery frequency rather than ground and they work on the "monitor ground" system and it has worked well.
The idea of doing the same thing with aircraft requesting pushback is a good one and has been looked at before and may need to be again. It is slightly trickier to change procedures like that with the computerised system Melbourne Tower has compared to Sydney Tower.
With the automation involved, currently as soon as an aircraft has correctly read back their airways clearance on delivery frequency, the electronic strip with all its details gets moved to the Ground Controllers console, and there's no way to stop it doing this. Hence, if an aircraft was to call back on delivery when they are ready to push back, there's no way for that controller to annotate on the strip that the aircraft is actually ready for pushback as they no longer have jurisdiction of that electronic strip.
With Sydney Tower's paper strips, the Delivery controller or Coordinator can hold onto the paper strip until the aircraft calls back and then physically pass the strip to the Ground Controller when they're ready - so much for technology hey!?
I'm certain this issue is not insurmountable and we can look at what can be done to modify the system to allow it to occur.