Most cars don't tell you how many gallons are left in the car, just approx how much of a full tank. So mpg doesn't help much in your example, miles per full tank is a useful number though and from there you can calculate that if you have half tank you only have 300/2=150 miles left. Though most cars will tell you the exact remaining distance to empty in the instrument cluster already so there is no need to do this mental arithmetic anyway.
As the owner of a very basic 2003 car, I can assure you that the mental arithmetic remains. I know that each 8th of my tank on the gauge is worth ~50 miles, entirely dependent on how efficiently I'm driving.
If I get 25 mpg and have a 12 gallon tank, that's 300 miles per tank. The nice thing about this is that it's a fixed value with simple, understandable units that more or less doesn't change as long as I keep the same car, presumably for many years. It helps me plan trips—I know that I regularly visit family who live 110 miles away, so anything with less than 220 miles to the tank forces me to refuel any time I see them… which affects my plans. America being less dense than Europe might partially explain why mpg is so popular here. A trip of 110 miles (or 200, 300) is not unusual in America. I remember making a series of trips to pick up a family member in college some 250 miles away, and that’s because she went to one of the closer options!
The same problem (averaging reciprocals) shows up in investment. If you get 10% growth this year and 10% decline next year, the average is not (1.1 + 0.9)/2 = 1.0 = 0% growth, it is 2/(1/1.1 + 1/0.9) = 0.99 = 1% decline.