All of the energy going into this should instead be directed toward "how do we vaccinate most efficiently in future?"
I think it makes a lot of sense to vary eligibility criteria by vaccination site, and allow for anyone to be vaccinated at some of the sites. That way, anyone who meets special criteria can go to a site where the wait will be shorter, and anyone who doesn't meet any particular criteria still have a place where they can go to get it provided they are willing to endure the potentially long wait. As the rollout progresses, if some of the "special criteria" sites are not too busy, they can change to become "anyone" sites.
Overall, the problem of "too many people want the vaccine ASAP" is a great problem to have.
I think it makes a lot of sense to vary eligibility criteria by vaccination site, and allow for anyone to be vaccinated at some of the sites. That way, anyone who meets special criteria can go to a site where the wait will be shorter, and anyone who doesn't meet any particular criteria still have a place where they can go to get it provided they are willing to endure the potentially long wait. As the rollout progresses, if some of the "special criteria" sites are not too busy, they can change to become "anyone" sites.
Overall, the problem of "too many people want the vaccine ASAP" is a great problem to have.