In the old days, medical expenses were more deductible (you could deduct the amount over 3% of gross income -- now it will be 10%). So people would pay cash, save their bills, add them up at tax time, and that would be that.
If the feds really wanted to simplify things, people would be allowed to pay for routine medical bills with their credit/debit cards, the provider would transmit a code to the card company that this is a deductible medical expense, the card companies would summarize the total and transmit a yearly 1098-type form to the card holder and the IRS, and the individual would just deduct it. BANG, no insurance overhead for the doctor, lower costs to the individual, less paperwork all around.
People could then go back to just needing to have catastrophic hospitalization insurance.
That would be too simple! Next Doctors would go back to making home visits and accepting payment based on what folks could afford, like eggs or produce. Oh the humanity!