Mark Cohen, I used noted harmonics to tune my guitars for many years. Two things caused me to search for another method: one was the problem of getting the B and G strings in proper relation and the other was the difficulty of tuning in a noisy room between sets. Using the method I describe, I can hold the guitar so as to press my left ear against the side of the guitar while I tune. This makes it sound very loud. When I play two adjacent strings, the natural vibration of the strings creates quiet little harmonics that, if you listen carefully, may be used just as you use the noted harmonics. The difference with open strings is that there is nothing to distort the note except perhaps the force with which the string is played. You want to be sure your are tuning with the same plectrum force and attack as when you play.
FWIW, here is my own theory about the difference in tuning methods. Keep in mind that I have no competence in the physics of music so I could be way off base. When you play a noted harmonic on a string, there is very little visible movement of the string itself. This means the harmonic is a function of the string pitch at very low volumes. When you are actually playing, the strings vibrate in a much wider arc. This wider arc means the tension on the string---and thus the pitch---is slightly different than the same string played softly. Using my method, the harmonics you hear are a function of the actual pitch you hear while playing. Test this with your electronic tuner. Tune the string to a precise pitch playing softly and then play the string hard and see if pitch reading has changed. I'll be interested in the result.
Rick, I suspect that is what's happening with your bass string in dropped D. The string is less taught and, when played with any force, swings in a wider arc causing the sounded note to go sharp.