More money per dose might have increased the efforts of the vaccine producers to speed up production and given them more financial means to do so. However, this still would not have been the optimal solution.
This is a case, where governments should have directly funded the build-up of production capacity. We need production capacities at a level which is economically infeasibly from a vendors perspective. Internationally, we need to produce like 14 billion doses per year. This would mean that whole humanity could be vaccinated within a year. Which is what we need, if we want to erradicate Sars-cov-2. But if we manage to do that, most of those new plants would become redundant and thats exactly why the government needs to pay for them.
What I don't get is why the EU doesn't buy the vaccine patents from e.g. Pfizer (or failing that, buy Pfizer outright - it would still only cost a fraction of what the lockdowns and aid packages have cost) and release the patents into public domain, so all companies with the capability can produce the vaccine. If it could hasten the reopenings by just one month, it would still be a terrific ROI.
Because there are no companies around which can produce mRNA vaccine. The production process is very specific. Opening up the patents wouldn't help. There are cooperations with those production steps which are generic, like botteling of the vaccines. Fortunately, another factory for BioNTech (which are the people who are making the vaccine, Pfizer is just the large partner) opened up in Germany recently and is going to produce a significant amount of doses. But instead of 1 more like 10 of these factories should have been built.
This is a case, where governments should have directly funded the build-up of production capacity. We need production capacities at a level which is economically infeasibly from a vendors perspective. Internationally, we need to produce like 14 billion doses per year. This would mean that whole humanity could be vaccinated within a year. Which is what we need, if we want to erradicate Sars-cov-2. But if we manage to do that, most of those new plants would become redundant and thats exactly why the government needs to pay for them.