Great question. I’m not sure. I believe the government sets the prices - that would be in line with how they generally do things. For some things, I think the patient pays a set “co-pay” and for other things they pay a percent. I generally paid about $5 (500 yen or so) for getting dental cavities filled, but I had a CAT scan (iirc - it may have been some other big machine they stick you in) and don’t remember how much I paid, but it was not an amount that we had to worry about.
I had national health insurance in Japan. They have plenty of problems, some of which will / do affect their insurance, but in general while I was there, it was great. Why? Because the government does NOT try to manage it. They just pay the bills. We chose our own doctors and our own dentists. We paid a small fee for each visit, and they sent the bill to the government. (Or, however it worked - i didn’t look into all the details, just know that we didn’t have to pay nearly as much as is typical in the USA.) Of course, they did take about 10% out of my paycheck each month, iirc.
Just saying that the British national system, which I hear is (poorly) managed by the government and has some serious problems, is not the only system we could use for inspiration.
BTW, I still get new glasses when I visit Japan. About $100 to $300+ depending on the frames you choose, and the eye exam is included for free. This is without insurance (I got 2 new pairs last year, for about $400 total), and I’ve not lived or worked in Japan in over a decade.) Thinner lenses than in the USA, but perhaps not as safe. The store is called “Megane Ichiba”, and I believe it is a chain.
Point is though, that the glasses market in Japan is not as monopolized (or at least not as greedy) as the market in the USA.