Just to clarify, for Canada - (IANAL)
Canadian provinces have single-payer systems that cover just about everything not elective. There is generally no bill for such treatments. You go to the doctor, or get admitted to hospital, and get the necessary treatment. The only bills you have are things like ambulance ride, prescriptions (unless in the hospital as part of treatment), non-hospital dental, and some extras afterwards, like a fancy wheelchair. These are the sorts of things “health insurance” as an employer benefit might cover if you have good benefits. Some treatments like physiotherapy may have annual limits.
Workers’ Comp - every province also has a workers’ compensation plan. Pretty much every employer must participate. Rates are set by the category of work, so retail store payments would be a lot less than say, a big warehouse or a steel plant or a construction site. Some very large employers are in their own category, basically paying what their employees cost the system. The WCB’s (workers Compensation Boards) are generally pretty anal about ensuring all accidents even minor ones are reported. Big industrial employers tend to be anal about safety - or they quickly learn to be. WCB also has inspectors for industrial sites.
So this is the response to the OP - if you go to a doctor, one of the questions will be “did this happen at work?” even if it’s a pulled muscle or a sore back. If the answer is “yes” then the doctor bills the WCB instead of the provincial health plan. (The major exception to the “single payer” principle). If you and/or the employer did not report this then there will be a bureaucratic shitstorm over failure to report within 24 hours. No fun being a boss. All subsequent treatment for that complaint will be billed to the WCB so they can track the true cost of workplace injuries. WCB also pays lost wages and even disability pensions or partial pensions. The major benefit to the employer, is it precludes almost any lawsuits over the accident. (IIRC, unless it involves criminal negligence)
Depending on your political bent, and that of the current government, the WCB is either too lenient or too strict. I know of one fellow who cracked his ankle (at work) and then, according to coworkers, walked on it frequently to ensure he stayed off work collecting pay for over 8 months… Except for the time after the new contract kicked in, where he went back to work for a day before deciding he was too sore and went back home collecting compensation based the new contract rate. Of course, while you are off sick - some insurance schemes were badly written and for example, paid your mortgage or car payments while the person was also collecting 80% of their wage, tax free. (It’s insurance payment). He resisted as hard as he could coming back on a make-work project painting railings. Another woman I heard of “snagged her bra strap” on a fire extinguisher hook, causing no visible injuries except leaving her with a “painful deep tissue injury” that she collected on for 5 months. Horror stories abound.
I had a friend in college when Ontario first published the list of doctors who made over $100,000 in a year. (Mid-70’s - Back when that was serious bucks). He explained his dad wasn’t on the list because it only listed the health plan, and not WCB payments; his dad made more than half his billings to the WCB.
For other incidents - well, unless you want to set up the equivalent of death panels - say, “Darwin panels” - how do you determine what to bill back to a patient? Is skydiving or hang-gliding so dangerous you should be responsible for your own medical bills? How about skiing - notorious for broken limbs? how about skateboarding? How about just doing really stupid stuff, like trying to get the barbeque ready faster by pouring on more starter fluid? The thing is, people don’t set out to injure themselves. besides, the hard-of-thinking tend not to have enough to cover their medical bills anyway. So whatever happens, stupid or accidental or whatever, the health plan pays for anything not work related; and compensation pays for anything that happens on the jobsite.
That’s probably a major plus - there are a helluva lot less lawsuits in Canada, because nobody is looking for $300,000 to cover the medical bills for their broken leg. Auto insurance will cover third-party for some stuff like physiotherapy out of the hospital and lost wages - if you are not at fault (and the driver was sober). If their insurance won’t cover it and they are at fault, probably yours will.
I’ve had experience out west with a single-payer auto insurance. It’s just like real auto insurance, except much cheaper unless you own a minimally insured POS. Their adjuster decides whose at fault, and the not-at-fault does not pay a deductible. At fault, your rates go up for a few years. I heard one province went to no-fault, where the only dispute was the deductible and all decisions were made by the auto commission’s tribunal or something - “you get $20,000 for all future physiotherapy needed.” Even if it wasn’t your fault, unless you bought the premium coverage, they don’t pay your car rental. (This is all second hand, so not sure…)
So with or without single-payer auto insurance, there’s a lot less lawsuits. Ditto for negligence; I slip on your icy sidewalk - no medical bills to sue for, so you have to convince a judge that you were so badly injured you need to collect months or years of lost wages, pain and suffering, etc.