>>the job has fallen to overstretched, underresourced state and local governments
If they are incapable of doing the job, obstructionist in implementation, and redundant, eliminate all of them.
In Connecticut, they were liberals online crying that States like Florida were using existing places where you can get things like the flu shot, including retail pharmacies, doctors offices, etc. Distribute the vaccine by the train professionals that already work there and are close by to the people.
Of course, for whatever reason, that was unacceptable and had a piece of government mandated saying in a giant parking lot where you sit in line in your car for hours we need to get your shot