Yes, but your idea is based on frequency domain with harmonics signal = small level = less amplification. You treat signal separately, as if the amplifier would see carrier only (large amplification), then harmonics only (small amplification).
Exactly, it is the sum! So with your exponential amplifier, the amplifier gain will then depend on the time varying instantaneous voltage of carrier + harmonics. Example for a heavily distorted sine wave: make it look more like a square wave, by adding a lot of odd harmics. How can any theoretical amplifier make that a sine again?
(If you are not familiar with the theoretical concepts of time domain <> frequency domain, look for "fourier transform" examples)
Your amplifier would need to split signals into spectral components first, and amplify all of them separately. FvM and I understand that your idea is inspire by the spectrum analyzer display, where that frequency separation happens. But an RF amplifier sees all the spectral components added, and amplifies at any time the input voltage which is the sum of all these spectral components. Then there is no "small harmonics voltage", there is only one sum voltage that is carrier + distortion.