The issue with shooting instinctively is that not many people can agree on what it actually means. The basic definition is a moving target (pun intended) in archery discussions and without an accepted base to start from it's near impossible to build a framework for improvement...either from the standpoint of shooting or with regard to constructive/cooperative dialog. In a nutshell, if we can't define what it is we can't improve how to do it.
Here's my simple definition. Shooting instinctively means that consciously you use no part of the arrow or bow to assist yourself in directing the arrow to the mark. The tools for instinctive shooting involve focusing on the exact spot to hit (see the spot relative to the distance), and having good enough form to allow the mind to carry out directing the arrow based on what it gleans from what it sees. The arrow is 'sensed' in the same way we sense surroundings, but not applied. That's basically it. A very simple concept both in theory and practice. That's what it means to me, but not to most. I think most people include the arrow itself as an ok component of instinctive shooting recipes, some as a minor part and others as a key neccessity. Personally, I believe that once you include the arrow in the mix you are basically practicing a form of gap shooting (split vision is gap shooting). And there is no such thing (to me) as 'instinctive gapping'. You either apply the arrow during 'aiming', or you don't. Not that there's anything wrong with using the arrow for that purpose, it's just that if you do then aside from discussing form the 'how to improve my shooting' part would best be addressed as a gap shooting issue, not under instinctive shooting.
ChuckC and Producer made some good points above with regard to how instinctive shooting 'works', and it's limitations. It's not magic but it also isn't gap shooting...and there are some basic realities to it. I guess the best way I can offer to describe what it is, to someone unfamiliar, is to imagine what you do when you move a loaded-up forkfull of tasty food to your mouth while reading the newspaper. No need to calculate anythimg to get it to the right place, you just kind of know. Yes, more complicated and nuanced than that in practice, but not really in concept. Once you've done it as a baby for awhile (babies don't do much calculating either, lol) you just know what it is and how to make it happen. I think Terry once used the analogy of it being like driving a car...seeing the hood but mostly the road. To me that's more like gap shooting...because if you remove the 'hood' from instinctive shooting you can still hit (stay on) the 'road', so to speak.
Two things I would say are relative constants about instinctive. The first is that 1.) it's not really instinctive in the pure sense...like sleeping. The word is only a catch-all label. And 2.) it really is a short-range affair with regard to maintaining consistency (short range as in under, say, 25 yards for hunting). I feel ok in saying this after doing it for a long time with both compounds and stickbows. Done well, gap shooting is admittedly more consistent for long ranges and makes a good compromise for long range target shooting. Sight pins are even better than gapping. I'm not promoting any one method over the other (instinctive, gap or pins), just providing an observation based on some experience with all three.
Personally, I enjoy instinctive the most because for me it brings out the true joy of shooting. It is a method that is closer to imperfection than the others but that offers satisfaction no other style provides. And that's important, especially as I get into my autumn years and care more about the joy parts and less about the perfect parts, lol.
P.S. On a side note, it has become fashionable in some 'trad' circles to promote taking shots at game at increasingly longer ranges, simply because archers are gap shooting more and seeing better results on targets at longer ranges. To me that is one of the biggest disadvantages (and disservices) of the gapping style...the inability (refusal?) of archers to separate target sense from hunting sense. I think even well-known and skilled hunters of old (Hill, Bear, etc.) would today admit they made mistakes in this regard, i.e. on taking shots they shouldn't have taken. I hope that as a group we find our own built in common sense meters before it's too late.