if the crew just went and got a pushback clearance then the airport car by law would have has to move as he is now obstructing a aircraft which has a clearance on a taxiway.
ATC only control traffic on the manouevering area, which does not include aprons, therefore there is no such thing as a pushback clearance. They give approvals only, and traffic avoidance is upto the crew and ground personnel. If you want to push and there is a car in the way, it's not ATC's problem. They cannot force anyone to do anything. They haven't given a taxi clearance, as the aircraft is not ready to taxi, and is not yet on a taxiway anyway. If there was a car on the taxiway obstructing the aircraft, the ATC would not either not issue a clearance, or cancel the taxi clearance and wait for the problem to be sorted out. Safety is their job, not security.