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Teens voting isn't much to do with trust, as it's to do with responsibility. As a dependent, it is assumed that your parents vote counts for youngest intention. With a car at that age, it's assumed that your parents take out insurance for you, still having the responsibility largely on them. 
Not that age limits have to be comparable. A 16 year old can drive and be responsible on the road, but their parents are still most likely responsible for upkeep and insurance. Voting, it's not that they aren't responsible for that, it's that their parents are assumed to vote in their interest. Parents are also taxpayers and vote due to contributing in the country in that way.
The car centricity of the US is a major problem. It's easily solved, don't live where a car is a necessity, but people are taught car = freedom from their parents in many cases. When I graduated college, and this was back in 2005, I chose where to live based on how well you could function without a car. I was called a lunatic. >

I agree but

I understand that in most of the country you simply cannot live without a car, which is a bigger problem in of itself

This is a big part of it. We need cars where I live. There is no way to get by without one, unless you stay at home all the time. Even those that lose their license will drive to work because they'll lose their job if they don't. They just take the tickets if they get caught and accept that as the price of doing business. 
So there would have to be HUGE infrastructure changes before we could really crack down on it.