In simple terms sending a ready message is telling the flow people that the aircraft is fully boarded, all doors are closed, airbridge removed and tug in place if required so that the aircraft is indeed ready to go off blocks now
The ATC delay can be suddenly reduced to require the aircraft to push back immediately and reach the runway holding point in the defined taxi time for that airfield.
So I would say that to a certain degree, the crew are correct to ask you to remain attached.
However, as you say - your company does not have enough tugs to push back all non-restricted aircraft while leaving a tug parked with all those that are restricted and have sent a ready message.
Therefore there are a few work-rounds.
1. Your company, the operator and local ATS can discuss what the issues really are. There will be a defined taxi time and the earliest slot will be now plus the taxi time and ATC have the ability to let the aircraft take-off as much as 10 minutes after the slot time. This can at times give you some small amount of flexibility after coordinating with all concerned. i.e. you go to push another aircraft but if the new slot comes through you can either go back and do the push if not already in progress with the other aircraft. If you are in the middle of pushing the other aircraft then provided that you are not going to be more than 5 minutes getting back then it should work - but it requires both operator and ATC to agree ton this!!
2. You can offer a tow to a remote self manoeuvering stand or area where the crew can wait for the ready message without need for the tug. Watch out for APU running issues etc in this case
3. At many major airports - they vacate the stand on schedule and if they have to taxi elsewhere annd shut down again then tough. Otherwise it costs the company lots of money and inbound aircraft have to wait fpor stands which make a whole pain in the ass of the schedules....which in turn causes CTOT's !!!