I'm not clear on the purpose of visualisation here. Is it just to memorise the game?
Depending on what is meant, I would give two separate answers.
(1) When I was a beginner I went through a brief phase of forcing myself to memorise pro games. I had a very good memory but found this extraordinarily hard. Even when I succeeded I found it a complete waste of time and abandoned the idea ever since. But what was hard was nothing to do with the length of the game or whether it was corners, sides or centre. It was always the first few moves. If I gave up and peeked at those moves, I could rattle off most of the rest of the game.
As I say, I gave that up and never found any reason to try anything similar in any other field, until I took up Scottish country dancing. The routine there is that we learn a new dance by doing a walk-through. Typically there are just four units (8 measures each in a 32-bar dance). But I found the beginning of each dance almost impossible to remember, even seconds after doing it in the walk-through. But once the dance starts and we get over the initial hiccup, the rest is usually straightforward. At first I thought it was an age problem, but eventually I discovered it was normal, and there is apparently a name for this sort of problem in psychology. No idea what that name is though, and I find it hard even to imagine what a rational explanation might be.
That doesn't help anyone, except perhaps to give reassurance if it happens to you.
(2) Much more interesting, and I think Kirby's answer is in the same bailiwick, is what is happening in your intuition. People delude themselves into thinking they control their brain. Your brain controls you.
Simply from familiarity with learning josekis, certain patterns stamped themselves in my brain (patterns, note, not sequences) and they are still there even 50+ years later, despite virtual lack of active use. This is what I would consider as visualisation. It means I can look at a game and instinctively have some sort of knowledge about how this and that pattern came about and what might happen next, with little details such as "there's a possible ko in the corner." Because of the way I learnt go, these patterns have usually been entirely in the corner.
However, some years ago, while T Mark Hall and I were transcribing the games of Go Seigen, I came across a remark by Go to the effect that people wrongly tend to ignore the side areas of the board between the corner star point and the side star point. Originally as a joke, I developed that idea into Go Seigen Groups - a safe group formed there has the devastating power of turning that quadrant into a Chernobyl area where nothing else - typically the much more numerous stones of the opponent - can thrive. But just developing that joke made it stick in my brain, and I gradually realised it was a genuine concept, not a joke. That made me look at (i.e. visualise?) these side areas instinctively in every game.
In more recent times, I have been looking a lot at old Chinese games, and development of go theory there can be described (with some artistic licence) as a search for the ideal 9-3 points. That made me even more aware of the side areas.
Blow me, I turn again, very recently, to old Edo games and notice how they were obsessed with the sides, especially, in the Honinbo family, with the 9-3 points. Then it dawns on me that the Honinbos would be the only ones lucky enough to have access to old Chinese manuals. So I end up focusing on the sides even more.
Just a couple of days ago, this had a useful direct effect. I was looking at a game from about 1690, and there was a joseki that was wrongly applied, according to the commentary, because Black played the side wrongly. Because of own recent obsession with the sides, my intuition recalled for me a game with a similar joseki but used properly in Genjo-Chitoku, a few decades later. In neither case were the sides specifically mentioned in the commentaries, but my brain had made the connection for me.
If that is what is meant by visualisation, it seems valuable, and can be developed. But not by mere repetition. It needs "effortful practice" - i.e. thinking about it and actively hunting out examples.
A friend of mine has done something similar with what we call Shuwa's nobis. Other players use them but Shuwa excels, and it is he who found a famous one in the famous ghost moves game by Jowa. Since Phil mentions Shuei, that brings to mind Shuei's famous L shapes, too.
To be specific, what I think is going on with this latter kind of visualisation is that, when you look at a new position, your subconscious brain brings up these sorts of moves for you to think about first. You "see" them before you see any other moves. I think we all do this with tesujis and tsumego moves, but it is possible to do it with other kinds of moves. I'd guess more people are now playing shoulder hits and contact plays because AI has focused their attention on such moves, so that these come up first in the visualisation queue.
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