Your comment got me thinking about some form of volunteer force of medics. Imagine if every 20th car that drove by had a government issued medical bag in the trunk and a driver guaranteed to have recent first aid and emergency training. And in return the driver gets some sort of tax break, or an HOV lane free pass, or whatever the appropriate incentive is. Surely some country has tried this out.
In Israel about one tenth of the population has undergone combat training, and I suppose about one twentyith of those are combat medics (including myself). We get no tax breaks or other incentives, but when the once-in-ten-years emergency pops up, we are quite appreciated.
From talking to other medics, I see that most of us carry simple useful tools, such as a cloth triangle, everywhere we go. And we're well-versed in improvising.
In many European countries, you won't get a driver's licence unless you pass a first aid test. And I think in some places it may also be required to have certain emergency equipment in your car.
The laws protecting the helper definitely help as well. As long as I'm not tremendously reckless or meanspirited I cannot be blamed (sued) for any damage I do in the course of providing first response aid. In fact I can be sued for not trying (calling for help is sufficient).
The US has so-called "Good Samaritan" laws as well. Some of them go so far as to protect untrained individuals who injure or kill someone while attempting to render aid.
That depends on your state. I took one for my first license but haven't had a driving test since when I've moved between states. I have had paper and oral exams since then, though. But that only shows I know the laws and rules of traffic, not the ability to drive.
Most first-world countries are not obsessed with self-elevation and interest resulting from their own infrastructure breaking down
Third-world countries might be more the incubator you're hoping for but they generally don't care what happens to the non wealthy (and the wealthy's FO plans there as with everywhere don't involve looting cars on interstates)
So if I were to guess, I'd go with no, no "other countries have tried this out".
E.g. in much of Europe you're not getting a driving licence without passing a first aid+CPR course, and every car must have a first aid kit and a fire extinguisher.
Also, I recall that USSR had mandatory civil defence courses in universities - so everyone with a college degree, no matter if it's medicine or math, will have had a full semester of disaster followup & related issues; didn't USA have something similar during the cold war?
There are search and rescue volunteers, and most counties have a team (that's usually roughly proportional in size to the population of the county), and while they don't get tax breaks, a few places are nice enough to offer them discounts here and there.
Medical training varies from team to team, but a minimum of wilderness first aid is pretty typical. EMT is pretty popular too.