That is a nice thought, fortunately NPR has done the actual reporting on this. It seems like the answer is government interference [1]:
"But the biggest reason German health costs are so much lower, experts say, is that doctors are paid less. This largely reflects Germany's concerted efforts to keep costs down over the past two decades."
On administrative costs it seems like government mandates on benefits and payment rates keep those costs 50% lower. Also, they have an employer-based system:
"On top of that, administrative costs are almost 50 percent lower. That's not because the German health system is simple and streamlined. With its employer-based system, multiple insurers and ever-changing rules, German health care is as complicated in many ways as the U.S. system. But administration is much simpler because nearly everybody gets the same benefits, payment rates are uniform and virtually everybody is covered."
==Most of the differences in outcomes between us and other wealthy nations amount to lifestyle/culture much more-so than affordability and availability.==
You have provided no evidence to back up this claim. What makes you so certain that it's true?
==Not having to pay a copay to visit the doctor is not going to change that.==
Might this encourage more people to visit health providers more frequently? This might lead to earlier detection/treatment and lower long-term costs. Why have you dismissed this potential outcome?
[1] https://www.npr.org/templates/story/story.php?storyId=919310...