1.) For all practical purposes, it is illegal to compete with OHIP (or wtv) for essential medical services. The few theoretical legal loopholes are totally impractical. In every way that matters, we ban private payment of health bills and the ability of health care providers to charge what they deem necessary.
2.) The "public" vs. "private" thing can be exaggerated. The issue isn't to have purely public or private healthcare, neither of which has ever existed, but to maximize health care delivery by best using each to its potential while also maintaining certain standards of social justice. As an example, public funding for optometry could be reintroduced using private insurance plans. It's not a zero sum game between the forces of public and private
3.) The actual insurance dimensions are unlikely to achieve any significant improvements in efficiency. More debate needs to be given to improving health care delivery as opposed to insurance. In Canada it's hard to do, given the decentralized nature of our health system. As an example, we know that the cost of a hip replacement can be a quarter of the cost domestically in medical tourism countries (i.e. Singapore) and any system could exploit that, public or private. The amount of potential medical reforms which could lower costs yet have nothing to do with insurance systems is staggering.
4.) Regulating fees may lead to inefficiencies. It's basic economic theory that the supply curve slopes upwards, meaning price ceilings lead to conditions of undersupply. Public insurers could commit to providing funding for socially necessary treatment, while still allowing health care providers to charge extra for those who want it. Limiting access to health care for some doesn't make it more accessible for others, and our current issues regarding wait times for elective surgeries is mostly the result of adopting a zero-sum funding strategy.
5.) We generally need to move away from treating health care like a pay-as-you go system. I think most would agree that health care costs are going up with the geriatric boomer boom. We know a large chunk of society is going to stop earning an income at the same time they will incur much higher health costs. That nobody has suggested forcing these people to save money during healthy years when they are earning for the inevitable geriatric health care costs seems odd to me.
6.) For those under 30 (an arbitrary figure), we should look into moving them onto high deductible plans with mandatory health savings accounts. If you are a spry 20 year old entering the workforce, you simply don't cost very much for the health system. If you managed to budget about 4k a year for long term health costs, by the time you retired you would have a sizable ability to meet your health needs. Rather than simply transferring today's income to elderly boomers, we should start making sure they don't run into the same problem down the road.