The problem is that there is close to no public transportation in the US. For example, as a German living in Hamburg, there is just no need for me to get my driver's license. Public transportation serves me fine and gets me places much quicker and cheaper than driving myself. Buses and trains usually are not crowded despite this being the second largest city in the country, and reach pretty much everywhere. And if everything else fails, you can just rent a bicycle (it's free for up to 30 minutes and all you need to do is walk up to a terminal and push a few buttons) and just ride that to your destination.*
When I lived in the US - admittedly, in Southern Maryland - for a year, I had to rely on people driving me everywhere, for there was simply no alternative way other than walking, period. Also, nearly no one owns a bicycle. At least that is my impression. I am not trying to say the situation in the US is 'worse' per se, though - as acquiring a car seems to be much easier and simpler than it is here. What I am trying to say is that, for the lack of decent public transportation (other than school buses) especially outside the larger cities, it would be highly impractical, albeit safer, to bump up the driving age to 17 or 18.
* - disclaimer - it should be noted that this applies to Hamburg, specifically, not to all of Germany. However, as even most of the rural areas are in some way or another included in an alliance of local public transportation providers centered in the nearest metropolitan area, I would still say that public transportation is far more widespread.