The mistake the EU did with the vaccine rollout is that it was talking to long instead buying. Otherwise something that needs to be taken into consideration is the following.

Right now we have a global pandemic so all countries world wide are affected. And unless the whole world is vaccinated we won't get past it since the chance that mutations occur that all possible immune against our vaccines increases the more people get infected.

Currently there are 4 Producing Vaccine "producing" countries/groups

USA, UK, EU, Russia and China.

From these 5 groups only the EU and China are exporting vaccines. I am unsure about russia. And the Chinese vaccine is also not that widespread outside China. But I know that the EU even supplies Canada and Mexico with vaccines, while the USA and the UK have export bans. That is one of the reasons why vaccinations are slow in the EU in addition to being slow last year with ordering.

So while they have great numbers at home concerning the amount of people vaccinated. It is morally debatable whether or not that is the correct approach for a global problem.