I recently posted a recipe from my school cookery book on another thread. There was a!so a shortbread recipe with detailed notes that went over three pages in the same book.
I think the recipe is fairly simple - it really is the technique that makes the difference. Living abroad as I do, if I want shortbread, I have to make it myself. I replace some of the flour with semolina, which gives a nice texture.
According to my school days recipe, you should have warm hands and, on cold days, warm the bowl and the fat but don't let it become oily. The mixture requires a lot of kneading with those warm hands until it is pliable and not crumbly.
Again, checking my notes from 1967, you should not bake it until it's brown, but just pale fawn. It will be soft, but when cold, is hard enough to remove from tin.
IIRC the oven should not be too hot and the baking time is quite long compared to simple biscuits. If you are making it in a tray, divide the shapes with a sharp knife while still warm, then leave.to cool and set hard in the tin until cooled right down before removing. Sprinkle with caster sugar while still warm so that the sugar sticks better.