I tried doing the same ( PMDG 737 at YSSY Gate 33, I couldn't find an E33 ), and pushback works normally.
It's possible you set the GSX "Verbosity" slider too low, or the delay between waiting messages too high, so you don't see that GSX is asking you to remove a pushback constraint ?
In case of the PMDG 737, pushback won't start unless your turn on the APU, but if you suppressed the alerts from GSX, you won't know it's asking you to turn on the APU, so it won't do anything until you turn it on.