I was thinking about this, and I think I actually did manage to push-start my car a couple of times, by parking uphill. It was a 1972 Olds Vista Cruiser with automatic transmission, so it certainly wasn’t easy to push-start. I didn’t think you could push-start an automatic, but my wife swears we did it, and I remember doing it.
I’ve push started an automatic with a weak battery. You can’t do it with a totally dead battery. The automatic transmission won’t push hard enough on the transmission to turn it over, but it does help. You need to do both, roll in gear AND use starter. Together the two will start up your engine. Sometimes. I’ve done it before. It’s really not worth the risk(imo) because if it doesn’t work, your car is stranded. I’d rather leave it in it’s parking spot and jump start it.
The other thing is, you need to get moving pretty fast...like down a long hill.
Also, you need to disable the neutral start switch. Doing so on a modern car is probably impossible. actually, push starting anything with fuel injection is impossible as well, unless you have enough battery power to pressurize the fuel lines. And now that I’m thinking more about it, there is a lock on the gear selector lever on modern cars that won’t let you shift out of park without depressing the brake. If there’s no battery power at all, you may not be able to get it out of park.
I suppose it is theoretically possible to push start an automatic with a totally dead battery if you got moving fast enough...but I’ve never tried it and I’ve never heard of people doing it.