When you say "thoroughly and consistently along the length of each string", it sounds like you imply that the string need actual stretching along it's entire length. That's not the case. You're not trying to stretch the string per say, but seating it at its contact points. Strings don't continuously stretch and go down in pitch unless they're starting to fail. The whole point of "stretching" strings is not to elongate the string or anything like that, it's to make sure it's settled and seated around the tuner, nut, saddle etc.
So yeah, you still need to press down on the string on a floyd near the nut and the saddle. On a double locking system, it's a far more important step than string stretching. Even on a double locking system, the new strings will curve around the nut and saddle, it's true for all guitars.
I still think the stringstretcha is snake oil, since they claim things that simply aren't objectively true regarding the physics of strings. I think the only benefit of it is that it saves on tender fingers if you're doing numerous re-stringings at once.