Sounds like they're taking examples from other facilities that don't fit your criteria.
I believe that for the facilities on 5/5 or 5/10, they can say they won't give EA unless you're getting vaccinated during your EA days. Prevents you from getting recalled.
If they get it during the days they work, sick leave isn't unreasonable.
But for the facilities that are on the BWS schedule, there's only a short window when a vaccination could happen that doesn't affect the facility staffing. For them, it should be the controllers schedule the vaccine when they can and then any schedule gap is filled with OT to cover for the EA.