I mean, it probably depends on where you are, but at least here (Ireland) this isn't true. The contact tracing and vaccination apparatus was run largely with people borrowed from other civil service departments, or hired as contractors. When the mass vaccination capability was stood down last year, they went back to their normal work and it took a while to start it back up again for boosters.

Did your local government actually hire large numbers of new permanent employees to administer this? This is (a) rather unlikely and (b) very much something you can check.