I AGREE wholeheartedly. After two years of trying every day to produce and stabilize the first three OT on Bb1--G1 or so, for kicks I started blowing the MP+reed alone before assembling onto neck. The very day I got my first reasonable one octave scale on the MP (by larynx elevated, chords mid-apposed, mid-tongue-raised-forward-thrust tip-down spitball-launch voicing... never mind, trial and error and muscle memory is the only way!) I found 7 OT for the low fingerings, front E3 sounded decent, and alt. G3 and G#3 were almost reliable.Practicing on the mouthpiece alone and learning to play tunes and scales helps to develop control of the oral cavity which is essential to playing overtones and altissimo.
Now I play a M9 range with tunes and patterns on the MP every day for just a few minutes and my OT, alt. and tone on the tenor are markedly improved and intervals more reliable.
By the way, first OT on the MP+neck alone is simple, as is bending down a m3rd.