Ooh, lucky!This might be a local organization issue. For me (not that far north from you) We were given a list of all the local hospitals giving the injection, along with links to each sites automated booking systems. Then we could shop around amongst 4 hospitals to find the combo of time slot/distance/availiblity etc to pick.
We get a text message telling us where the vaccination will be and the number to phone to arrange time and date. No wonder there's only a take-up rate of 87% for over 70s.
Doubtless people could phone to arrange a different venue but there'll be more than a few people who will just think "Stuff it!".
Perhaps we just have too many vaccination centres in the area - at least 6 "big" ones - making selecting the most appropriate one just too much of an effort.