do the math
You can do the math to get a better understanding. The generalized signal can be described as
A(t) cos ( f(t) t + angle(t))
That is the amplitude, frequency, and phase are functions of time.
Approximate the amplifier as a polynomial a1 v(t) + a2 (v(t))^2 and so on. Make three signals by making just one of the variables in the first equation a variable and the rest constants. Put them individually through the polynomial. What you get out is the revealing thing.
When A(t) is a constant you get out just harmonics of f(t) which can be filtered. When A(t) is a variable you get extra sidebands on the carrier.
As far as real world amplifiers go, if the signal level is lower the a2 (v(t))^2 and higher terms are lower.
Since the DC power into an amplifier is voltage times current, swinging the output much less than the power supply level is inefficient.