So I suppose I need to adjust my expectation that "approach" refers to radar information for arriving aircraft....
Yes.
At Cambridge for example if you call "Cambridge approach" and they don't reply "Cambridge radar" then they can't see you because they aren't providing a radar service just now.
(If you really want the radar service I believe you can phone up in advance, giving enough notice, and ask for it, but I don't know what the cost is.)