I was really surprised when I couldn't push start my 1992 Miata. I had the thing rolling down a hill at like 15mph in first for at least 2 blocks, engine was spinning, but just refused to fire. Jump pack fired it right up. I know the battery was dead after I left the light on, but I figured for sure the alternator would make enough juice to fire up the injectors and ignition...
Some alternators ironically require electricity to make electricity. They don't have permanent magnets inside, but instead use electromagnets. So from a stone cold battery, if there's not enough power to get those electromagnets functional, you don't have a way of converting that rotational energy into electricity.
https://en.wikipedia.org/wiki/Alternator#By_excitation
I do wonder how much current that requires, though. In a pinch, could a duct-taped string of AAs be enough to get you going?
Funny you should mention the Miata, the author was the lead concept engineer behind the 1990 (NA) Miata.
Use second gear. I have a '96NA, and first gear can't perform a roll-start, but it catches just fine in second. I have no idea why that is, but I remember I was just about of hill when I discovered it.
It's because the wheels can turn the engine more easily and for longer in higher gears. This isn't intuitive until you realize that you've flipped the inputs and outputs of the transmission, and this inverts the gearing relationship. _Higher_ gears are better at multiplying relatively little input (wheelspin) into a lot of output (engine rotation).
You learn the same lesson (2nd gear starts) with motorcycles, which have much smaller batteries and fragile charging systems so the need to push-start is unfortunately common.