The reason he is correct is that harmonics will tune based on Just Intonation, but instruments like guitars and pianos use Equal Temperament.
http://en.wikipedia.org/wiki/Just_intonation
http://en.wikipedia.org/wiki/Equal_temperament
If you tune a guitar using harmonics of adjacent strings, starting with a single string, then the guitar will not even be in tune with itself. From the page on Just Intonation, you can get the ratios that would be found in harmonics:
5:4 = Tt (major third)
4:3 = Tts (perfect fourth)
So if you tune to A=440 on the second string of the guitar (which is two octaves below A4, and therefore 110Hz), using harmonics, you'll get this (rounded to 3 digits after the decimal):
E 82.5
A 110
D 146.667
G 195.556
B 244.444
E 325.925
Notice that the low E and high E are not even multiples of each other (2 octaves above should be 4x, or in this case, 82.5 x 4 = 330) The two E strings will be out of tune with each other (and so will the other non-adjacent strings, but by different amounts).
People still teach and use the harmonic method for tuning, as well as the 5th fret (or 4th fret, for G-B). It is certainly better than not tuning, and it's probably close enough most of the time. But it won't be as accurate as a quality chromatic tuner that works with Equal Temperament.
Now I wonder how long before I regret posting this...
