From my personal experience on the 320/1s I fly it's not the pushback speed that determines the off blocks time, its the Parking Brake.
Sneeky way to get an OTD, release the parkbrake for a couple of seconds whilst holding the a/c with the footbrake. After a few seconds re-apply the parkbrake and it usually records that as the off blocks time, and in our case automatically sends it via ACARS.
Maybe you have a different mod though.
Dreamland beat me to it, but he is correct all doors have to be closed, although not sure about the slides armed, we can do it without that.