This question has a short-run focus whereas the biggest problem is the absence of a long-run strategy. It was clear that vast amounts of vaccine would be needed but there was no serious plan to install the necessary capacity. It is natural that the rich countries would be the first vaccinated since they are the ones who develop the vaccines and they have the necessary infrastructure to deliver them. However, that is not a good reason for the supply being restricted at this time, one year after the first vaccines were approved. The US, under George W. Bush, set up a system whereby poor countries could attain HIV drugs at low cost. This could have been incorporated immediately into the vaccine plans of the rich countries. This would have required substantial investments in production capacity while the vaccine trials were underway. This would have to be underwritten by the rich countries because private companies would not have the resources nor incentives to invest in such large capacity.
Had this been done, along with developing delivery methods that could facilitate vaccinations in poor countries, Covid would be a far less serious problem today. Hindsight is 20-20 but given the enormous value to world welfare, this was a sensible approach even if we doubted the viability of the vaccines. The cost-benefit ratio was surely high enough to overcome any reasonable risk assessment in, say, June 2020.
The problem was that the US did not provide the necessary leadership to make this happen. Had it done so then the other rich countries would surely have contributed to this effort. Why was there no US leadership? The answer is, obviously, President Trump and his administration. It is shameful that the US did not assume the leadership role that would have made the current situation far better.