Airliners like Boeing 747 and even Airbus 380 are at times considered as personal jets for extremely wealthy people. Looking from the side, Concorde was ideal for this role: while not very big, it probably could provide luxury transportation for ten VIPs or thereabout, adding to that something that looks more valuable and more exceptional than just a "flying palace" - speed like no other. New supersonic aircraft like Aerion AS2 are often seen in this role. However, it looks like rich individuals only bought various bones of Concorde after retirement.
The fate of F-BVFD looks especially strange. As I understand, it has been retired very early, in airworthy condition, just for being a surplus aircraft. After standing for many years in the open air, it deteriorated and then has been scrapped. I cannot imagine it to cost a lot during these years of standing. Why did nobody try it for personal use? Also, some pre-production aircraft have been retired after logging not so many hours and were airworthy.
Surely, Concorde uses lots of fuel, but probably the A380 burns no much less if per flight (not per passenger). And it only could fly supersonic over sea, but was surely capable of the fastest possible subsonic flight over the land.