That's not nearly as helpful for most people. They don't think in terms like that as having an intuitive meaning simply because cars don't work that way. When people think of problems of this sort they frequently use other data about the reference classes to help them. If you alter the behavior of the reference class they will generate confusion. For example, that's why so many people have trouble with "If some doctors are men, and some men are tall, does it follow that some doctors are tall?" but don't have trouble with "If some US Presidents were Republican and some Republicans were women, does it follow that some US Presidents were women?" Don't underestimate how much people use the actual behavior of a class to guide them rather than just the section of it that you have abstracted.

"Miles per gallon"/"kilometers per litre" and their inverses are more convenient when people will be integrating that information with "dollars per gallon" / "Euros per litre" to get the information they really care about: how much they will be spending on fuel. "Gallons" / "Litres", though they are volumes, are being used abstractly to talk about an amount of gas, in the same way we could have abstractly used mass to refer to the same thing.

Distance-per-volume seems intuitively comprehensible to people and they can usually do the simple arithmetic to answer the simple questions "how much range do I get from this fill?" or "how much petrol do I need to go this far?" If you told people their fuel consumption in "square metres" they'd look at you like you were some sort of nerd. (I like it otherwise, so I'm likely some sort of nerd also.)

Thank you for that humorous insight. I am entertained by the knowledge that my car has a fuel efficiency of 0.0784 mm^2.