I grew up in Texas, but have spent most of my adult life in Germany. It's not that Germans are innately better drivers, it's that there's not the same level of cultural entitlement to a driver's license. Driving is a privilege, not a right. This causes them to take it more seriously.
For starters, driver education is taken a lot more seriously - it's not a one-semester elective in high school or something your parents pay $500 for you to do over a few weeks in the summer before you turn 16, and you cannot take a road test without it, no matter how old you are. People save up for driver's ed in Germany; depending on how many lessons it takes for you to learn the actual driving part, it costs anywhere from 2000 EUR to 5000 EUR. Your license will have a note if you took your test on an automatic, restricting you from driving a manual shift, so everyone makes sure to learn how to drive a manual shift for the test.
They also more readily accept strict suspensions for a level of traffic tickets that most Americans would find excessively harsh - get a few 15-20 km/h (10-15 mph) over within a two or three year period, and your license will be fully suspended for a month, no "work and school" exception.
DUI is also taken far more seriously - if your license is suspended for that, there aren't any "work and school" exceptions either, and if you were drunk enough, or it was a repeat offence, you might have to pass the "medical-psychological exam" (MPU) to ever get it back, involving six months without touching alcohol and a bunch of other things that I've heard are a huge pain.
Part of what sustains widespread acceptance to high barriers to a license is that while Germans love to complain about how bad Deutsche Bahn (rail service) delays have gotten (even I'm starting to get irritated), it's still far more feasible to live a middle-class adult life without driving in a mid-sized city than it would be to in a comparable US metro area.
You'd also have to import German road design, construction and maintenance, and I'm pretty sure my people are unwilling to pay for that. The first time I visited home after a few months in Germany, I was initially afraid I'd get caught driving like I do here.
Nope, not even a temptation, because after a few months of driving here, the roads in Texas had too many random cracks and other inconsistencies for me to feel comfortable driving any faster than the other people on the road, and I even found myself driving a bit more slowly than a lot of the others!
I feel far safer driving here than I do in Texas or anywhere else in the US, no matter how fast the occasional vehicle blasts past in the left lane. The price of fuel and the level of strict attention that going any faster requires keeps most people cruising at a max of 130 kmh/80 mph.