If you're using a tuner, then you can safely use the octave harmonic to tune. She is wrong in saying blanketly that "the harmonics are slightly flat". Some are flat, some are sharp, some match equal temperament exactly*. This page has a figure that shows the relative sharpness and flatness of the first several harmonics.
Often people use "tuning with harmonics" on an instrument tuned in fourths usually means matching the "7th fret" harmonic of the higher string (a perfect fifth + an octave above the open string) with the "5th fret" harmonic of the lower string (2 octaves above the open string).
The problem is that the 7th fret harmonic is 2 cents sharper than an equal tempered fifth , so if you make these harmonics match then your open higher string will be 2 cents flatter than an equal tempered fourth above the lower string (or depending on your perspective, your lower string is 2 cents sharp).
If you start on the low E and tune all your strings in order, you end up with the G string being 6 cents flat (+- 3 human errors). If you start on the high G and tune them top down, the low E ends up 6 cents sharp (again +- 3 human errors). Perhaps this is a source of confusion that it relevant here.
*The asterisk is for inharmonicity, which is a problem with stringed instruments and not the harmonic series. Harmonics tend to go sharp, not flat, but the first few harmonics should be close enough to tune with.