It depends on the car and the problem. Sometimes you might push-start a car because of a malfunctioning component, like the starter or clutch safety switch. Sometimes the battery isn’t the issue, and if the battery isn’t an issue, you should be able to push-start the vehicle.
On my Civic, when the battery gets bad enough to need a jump start, I haven’t been able to roll-start it, but a tow truck driver was able to roll-start it when the clutch safety switch malfunctioned, and it would roll-start if the starter was broken. I think part of the issue is the size of the battery. My Civic has such a small battery that push-starting it with a dead battery doesn’t give any juice to the spark plugs.
Whether you drive a manual or an automatic, I’d recommend a good set of jumper cables, and I recommend jump-starting be your first option. Push-starting should probably be your last resort.