> they mentioned that "Air is free".
Water is free. Fetching water may cost.
Industrial Revolution started on water power. Once you have mechanical weavers, a water wheel is much more power than a bunch of laborers. And "free".
Well, the water does no work, the fall of water does work. The first mill on a stream could take water far uphill and dump it far downhill. When the inevitable neighbors (especially other mills) complained, water-drop rights were established. Your intake and outlet elevations were regulated. Compliance was enforced by your neighbors, nominally through courts, but sometimes dams and flues got the midnight axe.
> .... the engine which is driving the car can ...compress more air, to drive the car, which in turn compresses more air, which in turn drives the car, which in turn compresses... ... it will NEVER work.
When all available fall-of-water was tapped, but mills needed more power, they turned to steam engines. But steam engines were mostly adapted to pumping, not shafting; anyway "water is free" (and fall-of-water was already invested). So there were several installations where a steam pump picked up water from the tail-race and pumped it back up over the dam, while the wheel continued to drive the mill. Not a complete perpetual-motion loop, no. And wildly inefficient (but thermodynamic efficiency was not yet defined). And such a scheme can be used for storage: New Jersey and South Africa store electricity in ponds to even-out the day/night cycle of electric consumption.
> expanding air can be used to chill the car (free air-conditioning!) but skip over the fact that -in 'filling' the car, compressing all that air in a short fill-up will likely HEAT the air tanks to a couple of hundred degrees...
Filling should be done like any non-toy air compressor. Massive fins (even water-sinking) between compressor and storage tank. And vehicle filling is intermittent, while compression is probably cheapest if done slow and steady. So the local Esso-Air station would have a compressor running all night to bring tanks up to 5,000psi, then drop to 4,400psi in the morning rush, catch-up mid-day, droop again in the evening.... most air to vehicle tank will be several hours old and cooled simply by storage, on top of probably massive initial cooling to protect station lines and tanks and to maximize tank fill.
Expanding air will chill. We don't want to just waste it. IIRC, the better air-motors were "compound", high and low pressure cylinders. Expanding 300:1 can't be done well in a single cylinder. But also: expand half-way, say 17:1, and the air is COLD. Go a lot further, and it freezes. So there was an "inter-cooler" between HP and LP cylinders, actually to re-warm the air to recover pressure and avoid ice-up. So there is coolth to waste. But maximum at max power, and passenger cars do not run max-power most of the time. And OTOH, in winter this coolth is a problem, possibly requiring massive "radiators" for best efficiency.
> the compressed air car promises (regenerative braking), but electric cars can do in practice
This requires "downshifting", same as "engine braking". You want to pump energy from road to storage, different "gear" than running energy from storage to road. Electric motors have a number of tricks, from series/parallel windings to switcher-technology, to exchange voltage/current and force electric either way. Engine tricks are different, and perhaps not so elegant.
In general, for me, regen braking is useless. I run highway speed non-stop. All the energy I put into momentum when starting is scrubbed-away by air-drag (bearing losses are near-zero, and tire loss is small compared to highway air-drag). The most I could recover is the last minute before my exit. And I usually use very little dissipative-braking for that: I know I am getting off, I slack-off and spend my momentum to carry me off the ramp. There is a hard 25MPH-zero stop at the bottom, and that's the zone where regen braking needs impractical gear-up, usually resorts to friction to take the last few bits of momentum off. But I may be anomalous: I replace brake pads not for wear-out but because calipers suck and either stick or wear-off one corner of the pads. I see many drivers wear brakes harder than I do, and some of them stupidly excessively. A spurt-brake-spurt-brake driver could get considerable benefit from regen.