Golf is such a multi-faceted game with each hole and even each shot being so individual, mathematical models just can't be made.
Statistics can only set out to prove what they set out to prove.
GIR? An isolated event?
Putts/GIR? many factors affect
Sand saves? ditto
I'll stop now.
I have a couple of suggestions, one is a nitpick and the other a conceptual misunderstanding.
First, the nitpick. Statistics can neither prove nor disprove anything. A statistician never sets out to prove anything. Statistics is a way of summarizing and interpreting evidence to see if it supports a given thesis or not. Supporting is not the same as proving, not supporting is not the same as disproving.
Now the misunderstanding. When you pick a small set of useless pseudo-statistics like GIR, putts per GIR and sand save percentage of course you can't understand anything about the game no matter how you combine or formulate them. They are not informative.
It's a logical fallacy to say that since no combination of those statistics produce a useful model then no useful model of the game of golf can be made. Go collect me some real data and I'll build you a real model. You may ask what I consider real data...
Q: What is the fundamental event in the game of golf?
A: The stroke.
Q: What sort of statistics would be useful for understanding golf?
A: Outcomes of each stroke.
Q: What determines a successful outcome for a stroke?
A: How far the ball ends up from the hole (in the hole is best) and in what kind of lie.
This line of reasoning leads me to want data on where each stroke during a round ends up, distance and lie. Examples of such data records would be: 1) an approach shot ended up ten feet from the hole on the green, 2) a drive ended up 140 yards from the hole in the thick rough, 3) a putt ended up one foot from the hole or 4) a chip shot was holed out.
By stringing together these records you know where a stroke went from and where it went to. Maybe one stroke was a drive that started out on a tee and ended up 180 yards from the hole in the fairway and then the next shot started 180 yards from the hole in the fairway and ended up twelve feet from the hole and then the next shot started out twelve feet from the hole on the green and ended up holed out.
The final thing that would be nice to know is what club was used for each stroke. Given the distance and lie before the stroke, the distance and lie after the stroke and the club used for each stroke in each round for each player, you'd have a dataset that would be very informative about how the game was played.
The PGA Tour uses volunteers to collect this data at Tour events but I don't know anyone with access to it. You can subscribe to ShotLink but that just gives you predigested summaries in a flashy multimedia package. I don't think it is straightforward to get the data you'd need to actually do golf statistics. Playing around with GIR, putts, etc. is fun but it isn't going to tell you much that you don't already know.