Jon,
There will always be anecdotal examples of situations where someone had a bad day with their long game, but still shot a reasonable score because their short game was basically perfect that day. That's not a reliable way to play the game though. It might happen once, but it isn't going to happen every time out.
Mark Broadie did a whole bunch of research into this. His paper can be found here:
http://www.columbia.edu/~mnb2/broadie/Assets/strokes_gained_pga_broadie_20110408.pdfThe general story is that he looked at shotlink stats over an 8 year period (2003 to 2010) and over that period he took the average number of shots taken to complete a hole from basically everywhere. Then he measured how each player did relative to that average. From that he was able to tell where each player differentiated themselves from everyone else and by how many shots. Put that little lot together and you get his paper. Page 18 has some interesting numbers on it. Tiger Woods was clearly the best player over that period of time. 2/3 of that advantage that he had came from his long game (shots outside of 100 yards).
It is also interesting to note on the table on page 18 that Luke Donald has the worst long game of the top 10 over that period. He was ranked 65th for long game. That's still pretty decent and he was 7th on short game and 9th on putting. Donald and Harrington are the only two players for whom the long game doesn't make up more than short game and putting combined. Harrington still has the long game as the largest of the three parts of the game. Of that top ten however, there are Vijay Singh, Adam Scott and Sergio Garcia who rank 195th, 201st and 220th respectively for putting.
Correspondingly, of the bottom 10 performers (I think there's a minimum number of rounds to be included), all ten of them have the long game as the largest negative. Only Blaine McCallister has a larger contribution from short and putting combined than long.
Of the top ten, 6 are in the top ten for long game, but only 3 for short game and 2 for putting.
Of the bottom ten, 5 are in the bottom ten for long game, but only 1 for short game and 2 for putting.
Looking further down, the best long game is Tiger Woods - he gains 2.08 strokes per round from his long game. Craig Perks loses 1.79 strokes per round from his long game. So that's 3.87 shots from best to worst on the PGA Tour from long game.
The best short game is Steve Stricker - he gains 0.69 shots per round from his short game (inside 100 yards remember). Jose Maria Olazabal is the best chipper. He gains 0.30 shots per round from his chipping. The worst short game I'm not sure about but Guy Boros is 292nd and is losing 0.41 shots per round, so that's 1.1 shots from short game from best to pretty much worst.
The best putter is David Frost - he gains 0.72 shots per round from his putting. The worst that is shown is David Gossett (295th place out of 299) with -0.61. That's a range of 1.33 shots from putting.
So the range of best to worst for each of long game, short game and putting is 3.87 shots from the long game, 1.1 shots from the short game and 1.33 shots from putting. What does that tell you?
I can't tell you why Tom Watson wasn't dominant later on. I suspect it's because of a combination of a few things. One - that a stellar short game going to a poor one is still going to be a big problem. I'm not saying long game is the only thing that makes a difference, just that it's the largest. Two - that he is mistaken. Three - that the courses are significantly harder than they were when he was younger. Four - that the other players are notably better than his competition when he was younger.