A generator can be operated in current source mode by forcing a constant torque. In commercial power generation, the turbine is run at constant speed, in order to produce constant voltage. Load variations require that fuel burning varies to keep constant speed. Thus a CVS requires operation at constant speed & variable torque. A CCS requires constant torque & variable speed.
The voltage would be simply the current value I, times the load resistance R. For a dead short, power is near zero and speed is minimal. As load resistance R increases, the same torque results in higher speed and higher voltage.
The reason it is never done this way is that of losses. I^2*R is much greater than V^2*G. With CV, the full voltage is always generated, but current is that of the demanded load, 35%, 51%, etc. With CC, the current is always generated at 100%, with voltage varying with loading. Losses are greater in this case.
In addition, the CV is stepped up w/ xfmrs to increase V & decrease I. Again, losses are cut this way. If high temperature superconductors ever become real & affordable, constant current might be realized.
However, CV has another advantage - constant frequency. CF makes synchronous motors run at a fixed speed, and that is useful. CC results in variable frequency, not bad, but synchronous operation of clocks would be lost. Make sense?
Claude