I'm a new poster, but thought I should add my two cents of experience. Back in the 90's I started dabbling in making horn bows. I made several horn bows from Gemsbok horn and one from water buffalo horn. All of my attempts were poorly glued together and did not hold up very long. Back then a guy named Jeffery Shmidt from the University of Wisconsin had made a video and was helping me learn how to craft horn bows. I told him off my string of failures and, God Bless him, he just mailed me a bow core for free. All I had to do was sinew back it and cover it with stained rawhide.
It was a 60 inch Tartar(Mongolian) bow. I was living in Iowa at the time and tried to deer hunt with it the first fall after making the bow. As I was walking a heavily used deer trail, a buck ran across the trail with two does following hot on his heels. As I raised the bow to shoot the buck was already gone but I drew half draw and let the arrow fly at the last doe. It was really a random luck shot, no skill involved. It was probably an irresponsible shot, but even at half draw the arrow killed the doe dead in her tracks. Just by random luck the arrowhead struck her in the neck bone and broke her neck. She fell instantly and had about 30 seconds of spasms before expiring.
It was a close range shot but I was impressed with the force that it hit the deer. I think it was about a 60 pound pull.
The three finger draw is really uncomfortable with such a severe angle on the string at full draw. I tried mastering the thumb draw for years but could never master the method. With the arrow on the right side of the handle, drawing right handed thumb draw one can not aim properly by sighting down the arrow with the right eye. For some reason they always shoot far to the right. Last year I got out my old bows and tried to relearn the method. After reading an article by a mongol woman who competes in the Nadaam archery tournament I finally figured out the accuracy trick.
Here is how to aim. Grasping the bow in left hand, thumb draw with the right. Hold bow vertically, it is important that you do not tilt the bow at an angle. Draw to an anchor point just like the three finger draw. To aim, you sight with your left eye past the left side of the handle. In other words you sight on the side opposite of the arrow. From there you have to experiment to figure the height of your shot. One final note about accuracy is that you must have your face turned fully to the target. Proper posture is neccesary. If the turn of your head is not consistent then the distance from your line of sight and line of arrow flight changes, affecting trajectory.
It is definitely not an instinctive shooting method, sighting and anchoring is part of the method.