Hi...
One non constant envelope modulated signal has information into its phase and amplitude (AM).
So if u use a non linear amp, like a class C amplifier
that works in saturation, the signal's envelope will be flatted and the information that was on the envelope lost. Another problem is that u will have in band intermodulation of your signal, losing again information. For this reasons u will need linear amplifiers, losing in efficiency vs non linear amps.
Regards
Alex_vettoriale