I think the stats would prove that the winners historically make up twice as many shots not putting as they do putting.
A few examples don't mean crap, but here's one that illustrates what I'm saying: Phil missed a six-footer for eagle on 13 after hitting out of the pine straw between two trees. He birdied the hole because of his ballstriking. Tiger in 1997 won with his ballstriking. Wedges to par fives, lots of looks at eagle and easy birdie putts, etc.
You're taking too focused a view. The winner at Augusta National not only hits a lot of greens, he hits a lot of greens to good spots, leaving him shorter, easier putts. So yes, he makes more putts… but if you put lots of players in those positions, they'll make more putts.
Scott, Bubba, Charl, Phil not the best putters (T102, 157, 102, 133 for the years they won in Strokes Gained Putting). Seriously, not a single one cracked the top 100.