Economics 101 often fails to convey the crucial fact about rationing: under any rationing system that isn't a pure random lottery, the only effect of rationing a good is to change the currency in which that good is priced. [1]
For example, I live in Canada. Our healthcare system is single-payer [2], which is isomorphic to saying that it is rationed. So it would not be accurate to say that your access to healthcare in Canada is priced in dollars. But it would be accurate to say that your access to healthcare in Canada is priced in your ability to navigate bureaucracy, the flexibility of your working hours (to snag last-minute appointments), and your family connections who are embedded in the healthcare system as administrators or physicians.
Depending on your personal values, you might consider it good or bad that access to healthcare in Canada is priced in these units. What makes dollars unique is that if you inject more dollars into the free market for a good, then ceteris paribus, you will cause more of that good to be produced. Whereas if you injected more bureaucratic know-how into the Canadian patient population, you'd merely cause more intense competition for the same fixed number of physician-hours.
That is: by repricing a good in non-dollar units, you've removed a lot of the flexibility that would have allowed your system to make more of the good if lots of people wanted it. There are situations when this might make sense to do, and there are legitimate debates to be had around it. But this is the tradeoff you're making when you ration.
People often mistakenly think that if the price of a good is opaque, then the good must not have a price. This is incorrect. What's more, if you believe it, then you're almost certainly getting over-charged. And very unfortunately, this mistake tends to be most common among less sophisticated folks — who are the very folks that rationing was intended to protect in the first place.
[1] And unfortunately, real human-managed bureaucracies are not reliably pure random lotteries, even when they are supposed to literally be pure random lotteries. See, e.g., https://en.wikipedia.org/wiki/1980_Pennsylvania_Lottery_scan...
[2] More accurately, big subsets of our healthcare system operates under single-payer. Many services like eye doctors, dentists, etc. are private.