You're right. That is a flaw with the free market. A rich person mildly wanting something will outweigh a poor person desperately wanting something. That said, the alternatives aren't much better. A wait list doesn't have any concept of neediness. self-reporting is unreliable (everyone will just say they really want it). A neediness assessment (ie. auditing each applicant's neediness) could theoretically work, but requires time/resources, and still has biases (eg. people with "visible" problems probably will look more needy than someone with hidden ones). Moreover, none of other distribution methods incentivize further supply. A sibling comment covers this:
>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.