I used to have some CO2 extinguishers. They were always empty after the kids discovered that they made dry ice.
Here is my uneducated thinking about car fires.
One question is if the extinguisher is going to be reachable if the car is burning. If reaching the extinguisher places you at risk of getting burned it is not worth fighting the fire. Sorry but true. If the the car is already burning then an extinguisher inside it is a hazard rather than a help, especially if reaching it places you in harms way. Not only does trying to get it place you at risk from the fire, the pressurized extinguishers are potential bombs.
I think the biggest fire hazard in a car is battery driven electrical and fuel fires, mostly from under the hood. In an accident wires are often shorted together from the impact and making sparks, sometimes even just heating to the flash point temperature without sparking. Disconnecting the battery can remove that source of ignition.
If you are early enough to be concerned about the condition of the upholstery in a car you should be able to extinguish the fire with your bottle of drinking water.
If it is past that time then you are already past being worried about how much of a mess the extinguisher makes, and are likely past the insurance company's write off point for damages to the car.
If there is a fuel tank rupture you are looking at a large amount of fuel and again the mess from the chemicals are not going to be much of a concern.
In the case of a fuel tank starting to burn I would be more concerned about making sure everybody was clear of it, not because I expect it to explode like in the movies, but because leaking fuel spreads and burns fast with a lot of heat. Ten gallons of gasoline running all over the road and flaming is definitely a job for professional firefighters and certainly not a job for panicked accident victims with a small extinguisher.
Safety first, get the people clear of a fire hazard then fight it if it makes sense, but do the first aid for the victims before worrying about saving the vehicle. It is better to let the vehicle burn and save a life instead if you need to choose.
...Now there are other reasons to have a small extinguisher handy in your car. It might be somebody else's car on fire, or the fire might not be a car at all.
So I think yes for a small extinguisher, but with some cautions.
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Does that about sum it up?
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May set off to explore without any sense of direction or how to return.