Why doesn't anyone record through their amps?

All of the great players consider their amp to be an essential part of their sound. These especially include the jazz players who make a great effort to find the "right" amp to get their signature tone. But this is not limited to jazz guys; BB King, for example, will only use a Fender Twin for recording and live performances. A true tonemaster would never allow a recording tech to alter his sound through the board any more than any serious pro musician would let somebody mess with their individual tone quality.