Push and Hold allows the airline to say that the flight "departed" on time, (ie
ushed back from stand on time).
Crew duty hours start at report.
EU261 kicks in with doors open at destination time and then compared to original Scheduled Time of Arrival (STA)
All that push and hold does is improve "OnTime" departure statistics, nothing more, although as pointed out above, it does free up a stand for an arriving aircraft.