I can see that this would work with octaves, but for the other overtones wouldn’t they be in a different temperament to the horn itself and therefore not in tune with each other?
The reason for doing this is to set your mouthpiece so the horn plays in tune with itself. Playing in tune with other instruments is then up to you and your voicing skills. I only check the first two overtones, B1 overblown to B2 and overblown to F#2. Bb is not in tune on many instruments, and C can also be a little wonky, so I just do this with B. Once I find the correct spot for my mouthpiece I just always put it there. I only very rarely have to move it, playing in tune is MY job, not the mouthpiece's.
My experience doing this led me to have to push in - I was using too tight an embouchure, which raised the pitch of the "short pipe" notes. Once I was able to play with less pressure on the reed, it was actually easier to play in tune - because the intervals on the instrument were in line with each other. Loosening up also improved my tone a lot. It does require that your embouchure be stronger than playing tight - you have to have your muscles at just the right level of tension, but it's not full tension, and that can get tiring quickly if you're not used to it.
JBT's approach to this was what he called "mouthpiece pitch" - the note you play on the mourhpiece alone. This is a tool to help you form the proper embouchure and set your voicing up correctly, and is a common practice among certain saxophone professors. I actually find it to be useless, whereas he thought of it as an important technique, especially for beginners.
I can play about a 9th range on an alto mouthpiece, just by varying voicing (throat and tongue) and a little embouchure change at the extremes of the range. That's just a flexibility exercise, and I think of it as a parlor trick. That said, if you can't play at least a 5th or so range on mouthpiece alone, just by varying your voicing, then you need to work on your embouchure and voicing technique. I would think of this kind of mouthpiece work as an indication of your flexibility, but actual work on voicing and embouchure should be done on the full instrument. In other words, don't bother with practicing on the mouthpiece alone.