I have a feeling that for a spell Woody Austin (who from memory may have led the Masters after the first round on about 14 occasions) was putting one-handed, while looking at the hole and pointing at the hole with his other hand. Golfers are crazy people.
To add - when I played with people I didn't play with often and was doing the heads up putting, it was always a few holes in that someone would say incredulously "wait - were you looking at the hole when you hit that?" Actually made me feel a little self conscious while doing it - in a "I wonder if they'll notice it this time" way
A serve in tennis is a better analogy I think. The ball is fairly static and you're hitting it with an implement of some description and if you don't hit it with the right part of the implement, it's not going to end well.
I've tried heads up putting a few times. I think I'm backwards from most. I find it much more useful on the shorter ones than longer putts. I'm better at starting it on line and speed is less likely to be off using it. On longer putts where speed is more important, I apparently judge speed by watching (at least peripherally) my putter head. If I'm not looking at it, I lose speed control, even though that's the bit that is allegedly improved by heads up putting.