I don't think the "capitalist" system would really have any effectiveness here. Think about people like me: I could easily afford to pay whatever it would cost to be vaccinated. But, it's largely pointless: I work at home and don't go outside except to get some groceries now and again. If I did get COVID-19, I'm at a very low risk of dying. I'm probably not going to spread it to anyone either; I live alone and would just stay in bed for the two weeks I was contagious. And, if I did die, it wouldn't really matter to society -- I'm not writing software to save people from COVID-19.
Thus, if the goal is to minimize the spread of COVID-19, there is no value to letting me enter the highest bid for the vaccine. By the time my money feeds back into the system to increase production, everyone will already be vaccinated. Naively applying a simplistic economic model from elementary school doesn't get results here.
Capitalism doesn't help in the case that the article is discussing, either. Older doctors tend to be paid more than younger doctors, and are also at higher risk of COVID-19 without taking into account how many COVID-19 patients they see. If we just charged a high price for the COVID-19 vaccine, the older doctors would easily be able to afford it and get vaccinated and the residents wouldn't be able to afford it -- exactly what's being complained about in the article. You don't get paid for each COVID patient you treat (caring for dying pandemic victims is not a high-margin business), so using ability to pay as the means of distributing the vaccine doesn't maximize its value to society. The problem that the designers of the distribution model faced was balancing risk from old age with risk from exposure to patients with COVID-19. They balanced it wrong, and over-weigthed old age and under-weighted daily exposure to people sick with COVID-19.
It's not about capitalism, socialism, or anarchy. It's about getting back to normal as quickly as possible.