You haven't rendered private health care "obsolete", you've just forced people to join your public system. Obviously people aren't going to pay for two health providers (unless the public system is completely inadequate). That does not imply that the private service is uncompetitive. There can be no competition when one service must find willing customers while the other takes its funding by force.
This is the same issue with toll roads vs. public roads. Imagine there are two similar roads leading from A to B, one a toll road and the other public. The toll road charges drivers $1 per 100 miles. The public road is free to use but costs every driver $5 per 100 miles in taxes. Which one do you think drivers will take? The more expensive public road, of course—because they're already paying for it, whereas the toll road would be an additional expense.
This is the same issue with toll roads vs. public roads. Imagine there are two similar roads leading from A to B, one a toll road and the other public. The toll road charges drivers $1 per 100 miles. The public road is free to use but costs every driver $5 per 100 miles in taxes. Which one do you think drivers will take? The more expensive public road, of course—because they're already paying for it, whereas the toll road would be an additional expense.