What is really overrated is the notion that you can tune and play most instruments accurately enough to tell the difference.

Unless your "instrument" permits independent "tuning" of each individual note each and every time you play it, it is virtually impossible to "play" in "natural tuning" unless you limit yourself to music with less than a very few "chords." Carried to the extremes advocated (eventually) by most people who propose your argument, you would need a different B in a G chord than the one you would play in a an E (or B) chord.

In "a capela" singing (i.e. Barber Shop Quartets) good close harmony is always to natural tunings - and a goal is to get the voices to "blend" to the natural chord. Small ensemble fretless strings can often achieve the same result. Theoretically, a slide trombone quartet might get it fairly easily.

If you have frets on your instrument, or "keys" or "holes," you're not likely to get there - even in one key. With "modern" instruments, with accurate note-to-note intonation, a quartet of instruments all in the same family such as four clarinets, four saxophones, four cornets/trumpets, can harmonize to "natural chords" briefly - if all the players are skilled enough to "lip" the individual notes together - but it requires a lot of rehearsal to get there.

With plucked, fretted, instruments, the "ideal" of playing in natural tunings is also complicated by the change in pitch that accompanies any change in dynamics. A plucked string starts at a slightly higher pitch than the pitch at which it "settles" during sustain. The "louder" you play, the greater the difference. It is very difficult to "see" this variation with most accurate (electronic) tuners, since virtually all of them incorporate a "hold" feature that keeps the indication for the "first pitch sensed." Normal tuners "lag" the instantaneous changes in pitch enough to make the variations invisible.

If you have access to a "rapid response" tuner, you will find that a "hard" pluck on a typical guitar string will start as much as "8 or 10 cents" higher than the pitch at which is settles. This is MUCH more than the difference between "natural" and "equal temperament" tuning for most of the notes you'll use. (This is also the reason why the player who tunes "softly on the sidelines" and then plays at super-dynamo-air-raid-siren level is always out of tune during the "performance.")

There are times when "natural" tuning is appropriate. You should use them then. Most of the time, equal tempered tuning is better than what you'll actually play. If it makes you feel good to "tweak things" once you're close, then by all means do so, but don't sweat the theory.

John