... so, why is almost ever private jet ever made designed with the engines mounted to the fuselage at the base of the tail, rather than any other possible configuration?
The reason for mounting engines where they do on private jets is nothing else but "center line thrust". One of the problems of small jets is called VMC! That is "minimum control airspeed"! With the engines mounted on the wings, they are much further apart as related to the center line of the aircraft, there fore, the VMC will be much higher on aircraft with wing mounted engines, vs center line mounted engines.
There is a requirement by the FAA, when certifying aircraft for flight, that they meet what is known as a "balanced field length". What this is, a jet acierates to a certain speed, loses one engine, then how far does it take to stop the aircraft!
This is very important as a lot of small towns and counties have runways at their airport, where a lot of jets CANNOT operate from due to this requirement, so manufactures of jet aircraft for "corporate" use, try to balance out load carrying ability, range and "balanced field length", of which VMC speed is very, very important.
And last but not least is the noise factor, which with wing mounted engines is much higher than center line engines mounted on the rear of the aircraft.
There are a lot of other reasons for mounting center line thrust engines, center of gravity considerations, length of fuselage and even how high they want the height of the interior of the passenger cabin, for reasons of passenger comfort.
