This has always been something like my thinking. It's not like throwing a baseball or shooting a basketball where you have the object in your hand. You have to make one object strike another perfectly, so you should look at the spot where you have to make that perfect strike.
But... I played in a men's club tourney recently where by far the best putter in our group was one of the first guys I'd seen in a long time who looks at the target throughout the putter stroke. He sunk a couple 20+ foot putts and hit a bunch of very long lag putts super close with this method. My putting's been quite bad recently, though I changed my focus/thought that round and putted pretty well, but watching him putt had me thinking maybe I should give it a try.
Care to elaborate? Which type is more likely to have success with looking at the target? Linear? And we're just talking about degrees of arc-y-ness, since basically no one is literally keeping the center of the putter head exactly above the target line throughout the back swing and into impact?
I think I get the concept on long putts, but not so much the Jordan Speith way on short putts. I have no idea what I'm talking about because I've never tried it, but I could see how for long putts, where feel of the speed is 1000 times more important than hitting your line perfectly, perhaps some people would be able to "feel" the proper speed, or weight, better looking at the target instead of the ball.
@mdl is right that its not quite the same as other sports where you have the ball in your hand. Certainly consistent contact would be tougher to make, but I wonder what you might gain in terms of finesse. Think about it the other way: how hard would it be to shoot a basketball or throw a horseshoe well if you chose to watch the ball/shoe throughout the swing instead of the target?
If I could extrapolate that back to golf (I probably can't ;)) then I could see how people who are able to repeat their stroke easily might benefit from having their focus on the target instead of the ball.
Since I started playing I have taken a 50/50 approach to this theory. When I am practicing I set up my 2 tees and a ball then work on not hitting the tees and my tempo with my eyes over the ball. Then when I go out and play I am comfortable with my setup and everything so I look at the hole or my target. It really seems to help most of the time.