skymarker....interesting idea. But I think the question is not how many shots are you leaking, but where are you leaking shots? Seems to me that if you take away putts, the only shots left would be chips? Are you double-chipping? Are you sometimes failing to get out of traps? Or chipping from one side of the green to the other? Or hitting a 40-yard approach shot 25 yards or something?
You say you can't get GIR to budge, but I have a few suggestions about how to work on that.
1) Pretend like there are no flagsticks. Most recreational golfers score better when they aim right for the center of every green when there is no obvious problem (like a large trap right in front). When there are large problems, aim around them. But FORGET ABOUT the location of the flag ENTIRELY.
2) When aiming at the "center of the green," don't actually aim at the green at all. Rather, pick out a distant point that is on the line you want. A tree or a feature of a house or something. Forget about the green COMPLETELY. Just try to hit the shot toward that distant target.
3) Keep track of whether you miss greens short, long, right, or left. If you miss a lot of greens SHORT, as many people do, get in the habit of taking one more club for every approach shot. If long, go one less. If you are consistently left, aim further right and see what happens, and so forth.
4) Think about holes backwards; in other words, from the green to the tee. For example, on a par 5 that you can't get to in two shots, think first about where you'd like to be hitting your approach shot from. Let's say that you're comfortable hitting from 110 yards out, and because of the bunker left front of the green, the ideal position would be right side of the fairway. From there, plan your tee shot and your second shot to maximize the chances that you'll get where you want to be for the approach, even if that means hitting 5-iron off the tee and then a 5-wood to the right side of the fairway. Your chances of parring the hole go up quite a bit when you conceptualize the hole in that fashion, rather than just blasting away with the driver off the tee.
Good luck.
As far as your "new stat" goes, I kind of think that if there was utility in a stat like that, somebody would have already figured it out and starting tracking it and talking about it.