You write:
"As far as systems common-mode failure sources are concerned, the only possibility that has occurred to me so far is that the fuel feed system allows both engines to be fed simultaneously from the same (centre) tank."
I fully understand that you are searching for a common-mode failure - so am I!
But I have a big problem finding a scenario in which your CTR tank theory fits.
If the SOP was followed, and I have no reason to believe otherwise, the CTR tank became empty LONG before arriving LHR (as already mentioned in many posts).
Any fuel left would imply double scavenge pump failure - very unlikely!
In any case the AAIB report states that there was an indicated fuel load of 10500 kg upon arrival - distributed between the 2 wing tanks (5100 kg and 5400 kg). Nothing is mentioned about fuel in the CTR tank.
Also "the flight was uneventful until the later stages of the approach"
With the information we have received (so far), I cannot see that the CTR tank played any part in the accident.