You are skipping the starter.
the starter cranks the flywheel fast enough to create the compression/ignition in the cylinders to get the engine going in its cycle.
you are skipping the ELECTRICALLY DRIVEN starter and MANUALLY driving the flywheel fast enough by rolling the car in N, (flywheel stays still) and then putting it in gear and dumping the clutch fast (pops the flywheel in to rotation, matching the cars speed.. enough to start the process like the starter would do.
You just have to make sure you depress the clutch again after you dump it.. or else the car dies from being at a near stop (can't maintain idle at that low of a speed) and you are back to square one (pushing the car in neutral to gain speed)
so its:
neutral..
push car..
hop in..
clutch in..
1st gear..
DUMP CLUTCH OUT THEN BACK IN..
car is running...
start driving like normal.
Can be done in "R"everse if you are pushing the car backwards, as well.