With a single payer system you could also price it into things like red meat and alcohol through a sales tax that goes to insurance.
As a casual bicycle user, you're far more likely to have a serious bicycle collision from being hit by a car driver.
If a cyclist gets hit by a car at any significant speed, he's likely seriously injured or dead, while the car driver is likely unscathed, at least physically.
If two cars collide, unless they are traveling at quite high speeds or are particularly unlucky, everyone will probably walk away, as they are surrounded by a steel structure that has been specifically engineered to protect vehicle occupants in the most common crash scenarios, in addition to the protection provided by safety belts and airbags. The cyclist has none of this.
If two bicycles collide, well it depends but scrapes and possibly broken bones are likely, along with possible concussion or worse head injury.
This is an infrastructure problem, not a bicycle problem. The infrastructure problem exists because places like the USA prioritize cars.
Even the two bicycle collision you describe is quite rare outside of competitive sports. People casually bicycling to work aren't going at such speeds, don't go closely in opposite directions, and really have no reason to do so.
Bicycling is very safe when you remove car interaction from the equation.
Check out this YT channel for more info backed by data: https://youtube.com/c/notjustbikes