Key issue IMO is the chosen piece; selecting one, at least to begin with, having identical grip geometry etc.
That issue has come up in classes in the past. I think some guns are more conducive than others in that regard, but only in the sense that some have terrible grip angles which afford terrible initial results more than the majority handguns that are perfectly acceptable and will work just fine for this type of skill.
Usually when it comes up in class, I'll just ask someone to "hand me a gun", and then demonstrate the same proficiency with that piece as my own. Even if I have shot a similiar piece before or even owned the same model, the various grips of wood and their thichness, synthetics, rubber, etc have a way of changing things on the same model.
My favorites would include the 1911, the BHP, most glocks, Smith K frames with [ particularly with 4 inch barrels for me which I started with shooting from the hip in 1973 ], the D frame colts, and the list gets pretty lengthy from there. Most practiced for me at the moment would be the K frame, 1911, g17's.
I've seen some incredibly fast sighted fire shooter using FSP in competitions as I'm sure others have. Their rounds downrange counts are usually very high, and their dedication to "getting there" long in the tooth.
The results we are seeing students make at the ITFTS classes, and which are reiterated by the students reviews for the most part, is that for them, in the place and time of experience, they become faster, and some very much faster, at putting rds on threat in one day or a couple of days over a weekend. The average shooter who carries a gun on the streets just plain gets faster with hits on threats.
I've seen very proficient fsp shooters get 30%+ faster, both from quick kill hip and from just below line of sight using threat focus skills. I've also seen shooters with little real time on the handguns who, given the time, could make good hits when they took enough time [ that time being way too slow to respond appropriately in a crisis situation of SD with a handgun ] get a lot faster. In both ends of the scale, the students confidence level in their own ability was increased exponentially so that they knew they could draw and fire accurately without having to "verify" the sights in any way.
The skills can be imparted, but one of the most important aspects of training students in threat focused skills is the increased confidence levels in their ability to point the gun and land rds where they need to go on demand without taking any time to verify the sights. Becoming one with their weapon of choice. Again, in a day or two.
The point should be made that when time is of the esssence or you die, you need to have the skills and the confidence to put rds on threats without hesitation. There are techniques that are faster in this regard and they can be learned in short time frames.
One can take the time to develop their fsp [ some form of sights verification ] skills, hone them over longer periods of time till they have that same speed by streadily reducing their sight verification times as well.
Both sighted fire and threat focused skills work. One may need both at some time depending on many sets of circumstances they could run into. My own thoughts run toward using the most proficient technique to end the problem presented. There's no right or wrong way, but their is the most expeditious way.
Most students have skills in the form of sights verification when they get to class. We give them the other side of the equation and end up really building their confidence in their abilities. Many then go on to report their sighted skills are better as well after the training. There are a few reasons for that, but I won't expound on them here and now.
Brownie
You have a good point with using the same pistol and getting used to it. I use a Kimber CDP and have their 22 conversion for it. Makes a handy muti purpose tool. K frames are IMO the handiest revolver to point i have a slew of them some collectors and some shooters. I liked the older straight barrel (no underlug) better for carrying and plinking. 
