Actually it's those pesky overtones that resulted in my becoming a "no third in the bass when it's the root chord" fanatic.
Almost everytime I've worked with someone who wants to improve their playing, the first thing I try to get them to do is cover all six strings while playing first position "C, F, D, and B7". The bass note "E" in a "C" chord will sound whether played or not and makes for a "muddy" sound. Likewise the 5th string "A" in an "F" chord (not to mention the open "E"!!) I ask them to fret the bass F# note in both "D" and "B7" (with the thumb if possible). I know Segovia would barf, but I doubt if he played much blues, jazz or ragtime.
Rick