Most regular textbooks (e.g., genki, tobita) unfold in a linear fashion. Each chapter introduces new grammar points, and the texts and audio include those new grammar points, plus the earlier ones. Are Kanzen Master books structured in the same way?
For instance, if I do chapter 1 on the grammar book, do all exercises on the other books (e.g., listening and reading) in the first chapter only include the grammar points of chapter 1?
No, not really. The Grammar books lump related grammar points together to help you make the fine distinctions required between sometimes very similar grammatical points (or, as SomeCallMeChris likes to call them, "vocabulary"). The reading/listening books don't build on this in any sort of linear fashion. Instead, they are ordered by the different sections and question types you are likely to encounter on the JLPT.
Ok, so how could I tackle these books? Would it be a good idea to do first the grammar book (not necessarily very accurately, but relatively fast), and then start alternating the other books with grammar review?
I like the 新完全マスター series a lot. I used them for N2 and I'm using them for N1. I didn't work through the N2 book cover to cover but really wish I did. I got through enough of it though to get a pretty good grammar score on the JLPT. I used other resources too. I got the N3 after 2 years of study (in America) and finally got the N2 3 years later and one of those years was in Japan. So learning the N2 grammar was a hodge podge mixed up affair.
I swore not to do that with N1. I'm pretty happy with my strategy. I agree with the guy above who compared doing these grammar books with watching dead skunks mate. It's awful. But using them just as a reference I don't think will work so well for me. I got a PDF of the book and I'm going through each chapter I made flashcards for all of the multiple choice exercises using partial screen shots and pasting them into Anki. And I included the grammar explanation for the grammar point on each card. Copy and pasted. Making a chapter isn't too time consuming if you break it down so you're not jumping all around the PDF file too much. I'm on chapter 20 now so I've made quite a bit of progress into the book.
I don't know if it's the same but I downloaded an N1/2 grammar deck from the Anki share decks and was using it but the explanations were really weak and ultimately I didn't find it helpful. I think the deck author made the deck after completing an actual JLPT grammar textbook and passing the JLPT but still not being able to use said grammar points on command in conversation. So while it was a very effective deck for him, as someone who hadn't done those initial steps, it wasn't too helpful for me. I recommend grinding through an actual grammar book, be it 新完全マスター or one of the others.
Only partly relevant, but it's worth knowing that the Nihongo no Mori N2/N1 grammar videos on Youtube cover largely the same ground as Kanzen Master.
They are decidedly more interestingʕ•ᴥ•ʔ
You may not want to replace KM but the videos certainly help to reinforce it.