8 posts in and no one bothered to point out that it appears the OP does not know what single-payer means. Based on the other posts in this thread, it would further appear that no one else does either.
Now if the question is "why can't universal health coverage and private insurance co-exist" then the answer is, of course, that there is no reason why they can't.
Here is a good primer explaining the differences. I think people on here know the difference so I don't understand your comment.
source:
International Health Systems for Single Payer Advocates
Health care systems in the Organization for Economic Cooperation and Development (OECD) countries primarily reflect three types of programs:1. In a single-payer national health insurance system, as demonstrated by Canada, Denmark, Norway, Australia, Taiwan and Sweden (1), health insurance is publicly administered and most physicians are in private practice. U.S. Medicare would be a single payer insurance system if it applied to everyone in the U.S.
2. Great Britain and Spain are among the OECD countries with national health services, in which salaried physicians predominate and hospitals are publicly owned and operated. The Veteran’s Administration would be a U.S. single payer national health service system if it applied to everyone in the U.S.
3. Highly regulated, universal, multi-payer health insurance systems are illustrated by countries like Germany and France, which have universal health insurance via non-profit “sickness funds” or “social insurance funds”. They also have a market for supplementary private insurance, or “gap” coverage, but this accounts for less than 5 percent of health expenditures in most nations.
Sickness or social insurance funds do not operate like insurance companies in the U.S.; they don’t market, cherry pick, set premiums or rates paid to providers, determine benefits, earn profits or have investors, etc. In most countries, sickness funds pay physicians and hospitals uniform rates that are negotiated annually (also known as an “all-payer” system). Princeton economist Uwe Reinhardt calls Switzerland’s “sickness funds” quasi-governmental agencies(2). In France, the overwhelming majority of the population is in a single non-profit fund, so many observers consider the French system “single payer.”
There is no model similar to sickness funds operating in the U.S. (3), although they are often confused with the Federal Employee Health Benefit Program (FEHBP), which is simply a group of for-profit private insurance plans with varying benefits, rules, regulations, providers, etc. The 1993 Clinton health plan was an attempt to regulate private insurance companies in the U.S. to behave more like sickness funds, but the insurance industry defeated it.