twocats My friends say that when you can do that, it is when you know are fully secure. My friend worked on the devilishly hard Rachmaninoff Corelli Variations a decade ago and he ran it for me (with score) to see if he should program it for a performance and he still knew the piece really well! We were both amazed at it.
By the way, I'm not saying that that kind of memorization can not be done, but that it may a) only be possible for some people at a high level, and b) not be an effective way to learn a large amount of repertoire.
John Browning, along with Rubinstein, Gieseking and Richter, seemed to rely very heavily on thorough, near-photographic recall of the score. His contemporaries who studied with Rosina Lhevinne could not do what he could do and were fairly jealous of him. I remember watching in the John Browning documentary that Browning did not make any mistakes and didn't even accidentally hit other hammers or neighboring notes when he played pieces. Many such pianists point out what works for them and talk about it as if it's the "right way", but it really depends on the person in question.
And I think it's pretty clear that, generally speaking, obsessiveness with perfection gets in the way of musicality because it takes brain power away from the moment. I do think I hear that in many of Browning's recordings, and he isn't the only artist from whom I hear that.
I have actually been able to play through entire pieces in my head, and the last piece I really did that for was the Pathetique sonata, that I could play through in my head visualizing my hands on the keyboard. That was the one performance I think I had where I didn't make any mistakes, from what I remember (first two movements of the Pathetique sonata). But constantly visualizing what came next during the performance, I think, resulted in a relatively bad performance even though I got the notes that neither me nor the audience was satisfied with.
I've had a rough past year and that ability to visualize seemingly went away to a significant extent, although it's hopefully coming back. But that didn't hamper me from being able to perform on several occasions -- although my accuracy was lower, I was able to carry the performances with musicality and sound production, and I improved a lot. I did consistently notice that my sound production was the best when I pretty much wasn't even thinking a bar ahead and just immersed in the moment; however, it is a really risky way to perform. I aim for something in between the two extremes nowadays, but if only it were as easy as just being able to play through the whole piece in your head.
The reason I push back is really because I think it's truly impossible for most people, and kind of problematic to suggest other people adopt this kind of perfectionism because I don't think it works like that in practice in general.