In my experience the problem is exacerbated by those that are housebound or with mobility issues, that don't make other arrangements to get the vaccine.
We used to get inundated with elderly patients in particular, with flu that didn't think it necessary or couldn't be bothered to be vaccinated, or didn't know that a community nurse would jab them at home if they asked the GP surgery.
Usually, by the time we got the 999, call it resulted in a hospital admission.