You are making sense and it is confusing. Basically, I teach (I am a flute player) all my beginners to use double thumb Bb. Everyone has their own opinion so others may say otherwise. And honestly there is no wrong or right in this.

I prefer double thumb because when they are playing in flat keys there are fewer mistakes being made with accidentally playing b natural. I do show them both fingerings, but explain why I have them use the double thumb key. I teach my students to look at the key signature and if there are flats use the double thumb for all their notes - high Gb is later taught and explained with my intermediate students. I also teach that A# is fingered with the lever and not 1st finger. This is a personal preference. My reasonings are 1. it lays under the fingers more easily for fast passages. 2. It teaches them to think of A# as a different note than Bb.

To me the importance is correct notes and easy fingering. If the fingering is easier why do it the hard way? The keys are there for a reason, don't make it harder than it has to be. Not all flute teachers think the same way and that is fine. The end result and what works best for the player is what is important.