I kind of see it as one particular aspect of a whole line of variations. In some sense it’s what’s being said in the first chapter that the tesujis help speed up reading
the attachments to kosumis come up a good bit, as forcing moves. If you can force the opponent to go in a certain direction in the centre or to the edge of the board, maybe you can capture if they run into an edge or another wall, or maybe you can get a wall of stones you can use for another purpose.
The cross cut part is kind of branch of this attachment, attach + hane is kind of natural to put pressure on the attachment stone (removing a liberty) and then whether you can get a net positive from the cross cut, like capturing cutting stones, capturing the whole group or causing some other large damage to surrounding groups is kind of the thing making it a tesuji.
I do feel like some of the cross cut problems aren’t exactly straight forward overall, but just that the capturing the cutting stones is a possible variation is kind of what the message is to be taken out of it, I think, given the chapter.
I don’t think a sample variation like
is particularly simple, because then you need to evaluate what also happened on the outside, white captured two stones and they’re probably alive, blacks corner got damaged and they only got a couple of stones in compensation and so on.
As an aside, from my own perspective I don’t feel like the chapter really gives you the idea as to why some of the moves are what they are.
The net kind of makes sense, the nose into loose ladders or nets and so on kind of makes sense. Particularly when the surrounding groups are reasonably solid and the inside group has low liberties.
The knights move tesuji for me still isn’t very clear as to when you’re supposed to know that it works. Again I guess when the liberties are low or there’s liberty shortages like
where when black tries to break out their liberties are going to start disappearing really fast because of the empty triangle and kosumi.
Even though in the case of this modified shape
the knights move is super strong, it’s just black has a bit more to work with, hane underneath and tigers mouth and other ideas to try make shape, the liberty shortage not being a severe problem.