Skykes and Fairbairn's views need to be understood relative to their time. They published Shooting to Live in 1942. In 1942, techniques for shooting pistols were still pretty much based in the old art of dueling pistols, with one-handed operation being standard.
The point of the post and video showing the thumb under the safety is valid, but Fairbairn and Sykes explicitly denied what you wrote that I quoted above. At the outset of their manual they very explicitly excluded dueling as a legitimate purpose for the handgun being either forbidden or declining in favor where it wasn't forbidden. Instead, they set forth a dichotomy between target shooting and combat. (Handgun hunting was a burgeoning novelty at the time that they were unlikely to have heard much of, bearing in mind that while the book was published in 1942, the author's experience from which they write stems primarily from the 1920's and 30's.)
Here are their criteria:
"circumstances which preclude the use of a better weapon than the pistol -- that is to say, when it is impractical to use a shot-gun, rifle or sub-machine gun."
This should not be overlooked. They understood the pistol as a one-hand gun because if it was at all practical, both hands would be better used on a better weapon. While more and more officers have started carrying M4 carbines in the last decade, most of the shooting affrays they're involved in don't call for the M4's longer range or anti-3A capability. On the other hand, in many cases, they would be better served with a sub-machine gun than a duty pistol. Either way, a handgun is promoted by Fairbairn and Sykes as a compromise for specific unfortunate circumstances:
"Often the only warning of what is about to take place is a suspicious movement of an opponent's hand... If you have to fire, your instinct will be to do so as quickly as possible, and you will probably do it with a bent arm, possibly even from the level of the hip. The whole affair may take place in a bad light or none at all, and that is precisely the moment when the policeman, at any rate, is most likely to meet trouble, since darkness favours the activities of the criminal.... Finally, you may find you have to shoot from some awkward position, not necessarily even while on your feet."
They go on to espouse "point-shooting" or instinctive as opposed to deliberate aim and justify this primarily due to the need for speed. Most training in recent decades has substituted the "Modern Method" developed from competitive target shooting, something which the authors did dismiss. (I am not personally making an argument one way or another. It should be evident that some evolution of the modern method has practically usurped point-shooting in most training).
"There is no time, for instance, to put yourself into some special stance or to align the sights of the pistol, and any attempt to do so places you at the mercy of a quicker opponent."
No Weaver or Isosceles here. Not even a sight picture. In those circumstances, the support hand does not give any advantage, but training with it does hinder the development of one-hand skills.
So no, it wasn't dueling that engendered the one-hand technique through the 20th century. But it was competition target shooting that started the two-handed methods.