I think the massively better UK response to COVID vaccine distribution shows some of the benefits of independence.
The UK is planning to forego the manufacturer's and the WHO's recommended time between doses in favour of getting the first dose to more people.
AFAIK nobody knows how that will affect efficacy and might render the entire vaccination program useless, all in the name of getting a higher number of people vaccinated so the headlines look better.
In the name of saving lives actually. Two people getting 90% protection beats one getting 95%.