No "clearance" required to transit an ATZ.
Correct if the atsu is AFIS or just A/G
Incorrect if ATC
No actually, also correct for an ATZ with ATC. The commander must "must obtain the permission of [ATC] to enable the flight to be conducted safely within the aerodrome traffic zone".
A "clearance" is a somewhat more formal contract with ATC. When used in it's purest sense (as in the US system) it promises some level of separation in return for agreeing to fly a specified trajectory. I would concede that in UK usage for class D, it has become little more than a "permission".
On that basis, Cambridge (who are normally pretty good at asking what service is required) should not provide a
clearance if you're on a Basic Service, though it would be entirely appropriate if you agree a Procedural Service with them.
the controller sounded annoyed that I had not called with regard to my (above ATZ) overflight much sooner
While you may be correct in your perception, I'd caution against reading too much into an ATCOs tone. All sorts of stuff can be going on to raise workload (and hackles) that you may not be aware of.