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Couldn´t agree more!
I´m a good example of the second paragraph. Last tournament (72 holes - 6700 yards, par 70, rating 72, fast greens) I lost 20 strokes against a pro.
Drive: 5 (i´m a short hitter)  
Appr: 1 
Short: 1
Putt: 13 (just holed one 15 footer outside 10 feet in 4 rounds)
You could say that i just need to improve my putting to compete on PGA tour.. but no.. That´s my stats on a 6700 yards course. On a 7500 yards course set up for a PGA venue with a rating of 76 my stroke gained Driving and Appr will rocket to the sky, above my poor putting stats. 
  

Calm down man. We're talking here, and until @Herkimer stops talking*, we're all still in it.
* Though that may have already occurred, who knows…?

@Herkimer That’s pretty straightforward no? You’re not reading nor making any effort to think reasonable about this. Why are you doing that? What’s your concern about putting be the Holy Grail of golf? 

Here is a way of looking at it. At the end of the week, the guy who wins probably had a good week putting. There is a lot of randomness in putting. A machine won't hole every putt even if you get hours to calibrate it over a 20 foot putt, it won't hole them all. People hole 20 foot putts, but they also miss them. They miss them a lot more than they hole them. Even the best. If they have a week where they hole more than their fair share they have a decent chance of winning. But no one holes more than their fair share every week. 
It's a lot more common for a player to have a day where they have a +4 strokes gained putting than it is to have a +4 strokes gained driving. But it's a lot more likely that the guy who averages +1 strokes gained driving will be at +1 strokes gained driving every week. That guy who does that and gets on a hot streak with the putter is going to win a lot more often than the guy who averages -1 strokes gained driving and gets hot with the putter from time to time. 
The reason that Tiger was so good for so long is because he was +0.7 driving and +2.5 approach every round. He could just putt average and be in contention, because he was beating everyone so hard from tee to green. Then when he had his good days on the greens, he'd obliterate the field. That wasn't because he was the best putter though. It's because he hit it close more than anyone else and gave himself the most opportunities. 
The problem with thinking back on your own rounds is that you remember the days when your putting was lights out, because those coincide with the low scores. But you can't putt lights out every time. 
Your putting on a given day dictates where in your typical range of scores you are going to wind up. Your long game dictates where that range of scores is. 
Take a scratch golfer. One who is a mediocre putter and pick anyone in the world to putt for them. They're not going to make it on the PGA Tour. Not a chance.

That's an fairly bad way of looking at that. You'll find that "counting" stats are often bad, and this is one of those times.
In Lowest Score Wins we defined something we call "Separation Value." It's the measure of a skill's potential to affect your score.
Putting has a fairly low Separation Value. From 20 feet, the game's best players (PGA Tour players) take about 1.86 putts. A scratch golfer takes… about 1.89. A guy who shoots 90 takes about 2.02. So, over 18 holes, a full round of golf, the guy who shoots 90 will lose a whopping 2.88 strokes to the PGA Tour player if you put them at 20 feet on every green. He's lost 15+ shots elsewhere!
Putting has a low Separation Value®. It's not where PGA Tour players separate themselves from other PGA Tour players, and it's not where PGA Tour players separate themselves from amateur golfers. It's the area (of the four: approach shots, driving, short game, putting) of the game that has the least separation, in fact. They go in the order I listed: the greatest separation occurs with approach shots, with driving second.
It makes sense. My grandma (before she died) could make a twenty-foot putt. She'll never flush a 4-iron from a hanging lie to 25' on two-tiered green over a water hazard from 225 out. PGA Tour players get to the PGA Tour because they're the best ball strikers.
Counting stats are bad. A PGA Tour pro taps in nine times per round. Let's say they shoot 72. Those nine tap-ins count for 12.5% of their strokes. Yet virtually anyone in the world could have faced those same putts and taken… nine putts, same as the PGA Tour player. No separation there.
Please read this and let me know what your answer is, @Herkimer.
I'm a better putter than all but about five players on the PGA Tour (and that's not a hypothetical), yet I don't have a chance of competing on the PGA Tour. Why? Because my ball striking is nowhere near good enough, even as a +1. And when I say "nowhere near" I mean NOWHERE. And yet I'm a pretty darn good ball striker, all things considered.
Putting is the area of lowest separation, and counting shots is one of the worst ways to assess the value of a shot.
I'm not going to debate the idea that being confident is a bad thing, but… this is just a bad way to look at that in two ways:
PGA Tour players who often putt for par don't last very long. You're better off trying to get near the green, not laying back or merely "putting the ball back into play." Shorter shots are easier shots, and proximity is important. The best players aren't putting themselves into position where they need to scramble and make a 10-footer for par very often. Those players don't last on the PGA Tour for very long.   This is where your perception isn't really going to line up with reality. Yes, he occasionally did that, but what Tiger did best was drive the ball well enough, but hit his approach shots phenomenally well. Look at the chart above: outside of one year, Tiger never finished lower than fourth in Strokes Gained approach shots.
The year he finished outside of fourth… he was fifth. Yet his short game went as low as 89 (twice) if you exclude the year he was 160th, and his putting was 18th, 21st, 91st, 49th, and 27th.
Look at how the numbers in the "Total" column compare to the numbers in the other columns. Tiger was first in strokes gained total 7 years, and second once. Because… of his ballstriking. When his driving suffered in 2010-11, he finished lower in total strokes gained.
This isn't atypical.
This is part of the problem: you're believing your own experiences or what you think someone else may have said over actual data.
The data isn't biased. It just "is."
Open your mind.
There's another topic that'll probably blow your mind.