Not just the quantity but also the logistics seems to be another problem for the vaccine rollout. If we continue to innovate and have all the countries do their thing and become competent, I am sure that we can make the inoculation much more faster, we can't do anything about it, if the government is incompetent with the vaccine rollout then we as their citizens are screwed.
Logistics is a problem because the mRNA vaccines need deep-freezing. The vaccine from Pfizer/BioNTech needs to be stored at −70ºC. The Sputnik V vaccine also needs to be stored at -18ºC. The developing nations may be having the facilities to store the vaccine at such low temperatures, but we can't expect the poorer nations to have tens of thousands of such deep-freezers. And it's not just storing. Transporting the vaccine at such low temperatures can also be a problem.