To expand on this a bit, think about how you play when you have been putting really poorly (which is relative to what you consider good and bad putting, poor putting could mean you don't make everything under 8 feet, or could mean you three putt eight times a round instead of only three)
My experience, and that of most people I've explained this concept to, is that when you are putting well, you tend to play your short game with less stress. You feel like your putting will save you if you hit a poor chip or sand shot, and you will still save par (or for an egregiously bad short game shot, save bogey instead of earning yourself a quick double) Because you are less worried about your short game, you tend to play more fearlessly on your approaches, and are less concerned about all those bad spots that normally would be on your mind because they mean bogey or worse for you. You're more likely to play at a tight pin and be less concerned about short siding yourself. It even feeds back into your tee shots. You are less likely to try to press it for an extra 20 yards hoping to play a 7 iron instead of a 5 to increase your odds of hitting the green. You are less worried about the consequences of a bad shot, because you feel like even if you knock it in the trees and have to lay up or punch out, you still have a decent shot at par. As a result, your tee shots are played with less stress and are thus more likely to have a good result.
Now think about how you play when you are putting badly. You put lots of pressure on yourself with every shot game shot to get the ball within a few feet, because you don't feel confident you'll make anything outside kick in range. On your approach shots you aren't concerned with just the really bad "you're dead there" spots to miss it, you don't want to miss it anywhere. You might even worry about aiming away from a sucker pin and don't want to aim too far away from it because you're worried if you leave it too far you might three putt. And of course off the tee you feel like you MUST get it in the fairway to have any chance at par, and you have to really blast the crap out of it to have even the tiniest prayer at a birdie.
Personally, I think this is exactly what went wrong with Tiger's game. You could see the first signs of his putting leaving him about 5 or 6 years ago, when he started to leave putts short on a regular basis for the first time. At first it was only on long putts, but soon it was even on 20 footers that Tiger used to think he could make every time. That's what happens when you lose confidence, you're afraid of being too bold and leaving a 3-4 foot comeback so you start lagging everything. Then he started missing short ones and the wheels start coming off his whole game. It wasn't just the drives way right into the trees, you'd see him hit short game shots that were not only mortal, they looked like they were hit by a guy who can't break 90.
So don't give me this crap that his dominance was in his long game, or that the long game is more important. If you don't have a short game, it doesn't matter how good your long game is, you can't play well on a consistent basis. Poor putting and/or short game puts way too much stress on the long game.