How about we look at something like Singapore has? A system funded by a payroll tax, where the revenues are places into individual accounts that can be pooled and shared with families, and the government covers 80% of the costs of procedures performed in public hospitals? The remaining cost can be covered with these mandated health savings accounts where employers are mandated to pay into, much like a 401K. The government can also provide a disaster relief insurance plan where you can pay for its premiums using your savings account, if you chose to do so.
No healthcare procedure is "free" like it is under some countries, but adopting this type of system would ensure people don't overspend on health services, which cause the cost of something like this to go through the roof.
There is also an option for private insurance and private hospitals under this system for those who wish to purchase it, and employers can offer private insurance to their employees should they chose to do so. The government sets the prices and policies for these private insurers too.
This idea appeals to me, but it is said that it could be difficult to replicate in other countries, so it may not work here.
The poor and the unemployed just would not be able to pay that 20%.