I apparently have some divergent views about this subject.
First, I don't think of it as trigger pull, I think of it as trigger press. Give the two words some thought, and I think you'll get my meaning without further explanation.
Second, I don't apply pressure until the sear breaks, I apply pressure until the trigger contacts its rearward stop. The latter ensures a more complete follow-through. I see shooters popping their heads up, as if to see something downrange (I can't imagine what) immediately as the shot is going off. This is exactly what proper follow-through is intended to prevent, and this also almost guarantees erratic dispersion. In fact, this behavior is at least as prevalent, and as harmful to accuracy, as a full blown flinch. I've seen it over and over in youth marksmanship instruction, and it is one of the behaviors we have to work hardest to correct.
Next, I think that the 'surprise me' mantra is overused. If a shooter is doing all the repetitive shooting and dry firing it actually takes in order to acquire a reasonable working muscle memory required to properly 'know your firearm', such a 'surprise' is essentially impossible to achieve. You know, no bones about it, just when that shot is gonna break. In my case, after competing with the same rifle/trigger for about a dozen years, it becomes an act of will to simply apply and complete the trigger press maneuver without allowing the body to act on its anticipation. The trigger release is the most difficult and nerve-wracking process I face every time I fire in comp.
If you can't call your shot, the odds are you're anticipating the break. As instructors, one of the things we watch most closely is the shooter's aiming eye. Without exception, when the eye closes that instant before the break, instead of the instant just after, the shooter is anticipating. From that point, it is essentially a given that the shooter is forcing the break, tensing, and throwing the shot wide of the POA.
Knowing when the break is imminent is the most destructive detriment to proper trigger release and followthrough. But assuming that we can always fool ourselves into believing we can engineer a surprise can be true for some, and fiction for others. This may be an inconvenient fact, but it is a fact nonetheless.
I shoot with support (sometimes a rest, sometimes a bipod) and a rear bag, invariably from a bench; my old bod cannot endure a complete course of fire in the prone anymore. I don't grasp the grip, but either pinch the trigger with forefinger and thumb against the rear of the trigger guard, or I rest my thumb immediately behind the receiver on the grip, centered left/right on the grip. Aside from a gentle/snug contact of shoulder and rifle butt, and a light cheek contact with a carefully adjusted cheek rest, these two digits are all that touches the rifle. The rifle is reset against the forward stop of the rest, or preloaded into the bipod, with the rear bag repositioned to the same relationship with the butt stock before commencing each shot check-through cycle. The intention is to try and ensure that the forces involved in operating the trigger are running parallel to the bore axis, and that each recoil cycle begins from the same relationship.
I am learning to use a new Caldwell Fire Control Rest, with a joystick position adjuster. It has some benefits, but it's not any kind of panacea, in case anyone was considering one. I like the joystick adjustment feature, but I don't count on it to do more than keep up with bag settling during the course of fire. The forward stop is the most beneficial new feature I've gained with with it.
It is only with consistency that a consistently repetitive POI can be established and maintained. When I can see shot after shot stacking, just slightly enlarging the same bullet hole at 250yd, I know that whoever that shooter is, they have mastered the competitive trigger break process.
Greg