Pianists differ pretty widely on memorization, I’ve noticed. There are many who will conscientiously memorize music a few bars at a time.

The process I’ve sort of fallen into over the years goes something like this: when I’m getting to know a piece, I’ll practice it with the music for quite some time before I allow myself to even think of memorizing it. When most of the technical issues are ironed out and I’m able to do run-throughs, at some point I’ll take the music away and see how much of it I can play. Almost always I really surprise myself, but often there are a few “spots” that I have to pretty carefully review. 

I would say the downside to the kind of memorization you’re describing (if I understand it right) is that it focuses on “muscle memory,” that is, memorizing movements instead of musical ideas. A certain amount of playing from memory is always going to involve muscle memory, but relying on it can create a situation in which you might find it very difficult to regain your composure, for instance, if you had to stop in the middle of a performance.

So I’d say it’s partially about connecting the page with your fingers, but mostly it’s a matter of internalizing the music as music and not as physical activity, though it certainly is that, too. It helps me to spend a lot of time just sitting and studying the score, listening to it in my head. But everyone learns and works differently, so there’s no right or wrong answer.