When I was young an dumb I drove a car with a dead starter to work and back for a week. I'd bump start it by pushing it up to the crown of the street.

I think a classic automatic transmission needs hydraulic pressure to engage the bands. And since the gear pump is on the input shaft side it doesn't run when the car is pushed or towed.

"The old Model A had a spark advance you could manipulate. I don't know why they got rid of it. Well, that's your Detroit smarties. The hand choke too. That's gone. Been gone." - Charles Portis, The Dog of the South, 1979

You can't even clutch start some modern manuals.