Such harmonics and overtones are captured by the microphones. The job of the amp is reproduce the captured signal as accurate as possible. It's NOT the job of the amp to reproduce additional harmonics. However, many amps, especially tube amps, add additional 2nd and 3rd harmonics, which is pleasant to the ears for many. *hint hint*
While I understand that some amps are more natural sounding than others, I don't think that's because such amp is capable of reproducing the harmonics more naturally. Most all amps have flat freq response, and are able to reproduce captured harmonics correctly. I think it's the added harmonics that the amps generate that influence the perception that one amp has more harmonics and therefore more natural sounding than others.
While I understand that some amps are more natural sounding than others, I don't think that's because such amp is capable of reproducing the harmonics more naturally. Most all amps have flat freq response, and are able to reproduce captured harmonics correctly. I think it's the added harmonics that the amps generate that influence the perception that one amp has more harmonics and therefore more natural sounding than others.