Why was Concorde never sold as a private business jet?
Private business jets have a few requirements:


Operational costs must be within budget. The concorde is pretty expensive to run as demonstrated by the high ticketprices you had to pay for a trip on one of them. Fuel is only part of that costs; maintenance is a pretty big part of that as well.
Be able to land and take off close to where the CEO has his meeting. This means making use of small airfields with their short runways and noise restrictions. Concorde needed a lot of runway and was not a quiet plane.
Account for airfields sometimes not having fuel available or the fuel being too expensive. This may mean doing the roundtrip on a single tank or doing another short hop to an airfield with fuel.


Business jets would also mostly operate on a domestic scale. Doing mostly short hops where the concorde would not even reach supersonic cruise altitude before it's over the destination. While plenty of business jets can easily do 0.75 mach which is already 3/4 the speed of sound.