Filling a 200 V capacitor from a 12 V battery
a 200 V capacitor with a small amperage

Capacitors do not have "amperage", they have "capacity".

Good capacitors (low ESR) are capable of delivering a high current (until it is discharged). How much current and how long depends on that capacity and ESR.

Sure, an AED could be built into every car. But having the 12 V available is not a sufficient reason. All houses have mains voltage, it can also be used to power an AED. The reason why we would not want to have an AED in every car is cost. The cost is not the power source, not the electronics to make 200 V from some battery voltage and also not the 200 V capacitor but the electronics needed to safely revive the patient !

Also, when you need an AED you want it right next to the patient. So an AED which can operate stand alone, I mean, no wires, uses its own power source, is much more useful. So AEDs almost without exception use non-rechargable Lithium batteries. These have a long shelf-life (hold their charge for a long time) so they will work when they're needed. You want that when people's lives depend on this. With a Lithium battery it would only need to be replaced every 5 years or so.

An AED is very rarely used so that 5 years is needed to guarantee that that battery is OK, not to power a 200 V capacitor. Charging a capacitor takes less than a minute. Do you have a camera with a flash ? Do you sometimes need to wait for the flash to charge ? There's also a 200-300 V capacitor in the camera which needs to be charged. Keeping it at 200 V all the time is not energy efficient and not needed.