It's not like a grammar book. It's more like vocabulary. And the vocabulary is using very specific examples that you're not likely to use in every day speech.
If I think of grammar in the martial arts, I would be thinking of things like where techniques go and how you apply them. Grammar is that you capitalize the first word or proper nouns. You put a period/full stop between sentences. You put quotation marks around quotes.
Its more than vocabulary words. Why would a vocabulary list have the same word multiple times?
The first kata in Shotokan shows the front stance. It would make sense to teach beginners the basic stance of your art, in the first kata. Stance and foot work have a lot to do with martial arts. You then learn basic movement in that stance. This is about controlling your body, and learning to move explosively while in control and on balance. You learn to generate power from the ground, through explosive movement of your body, and connect it to your attacking strike.
It starts with a very basic strategy... block and counter attack. In the same kata, this strategy can be evolved over time for the more advanced student, to shorten the gap between block and strike, to look at how the first movement aids and supports the second movement. Note that the beginner and the advanced student can be doing the same moves, at the same time, focusing on these two ideas independently. Later, that block can become a movement to clear the opponents guard and create an opening for the following strike. Then the block can become a strike, followed by a strike.
This kata also gets into showing a circular move followed by a linear move. Circle defeats linear and linear defeats circular. Then it sets the pattern, only to then break the pattern.
We also get to explore breathing in this first kata. When to breath and how to breath. How to connect our breathing with our movement and with our attacks. What happens to most people the first time they spar? They forget to breath and gas themselves. The beginning lessons are here, on how and when to breath, while moving, attacking and defending.
This kata also starts to introduce general strategies. Entering in. When the attack comes, you enter in and take the center line. That block you make before the lunge punch, should not only block, but should off balance and destroy the structure of your opponent, allowing you to enter and own the center line.
These are all ideas that can be explored here, with these basic moves. But these ideas, tactics and strategies can be applied with any number of techniques... as your vocabulary grows. I am sorry that you don't see these same type of things in TKD. When I look at the beginner level forms from TKD, I see many of these same things... though some are expressed in slightly different words. And as you point out, they are coming from a slightly different point of view, through another culture. But I don't believe that watered anything down, or took anything away. If anything, it may have added to the depth or at least had a slightly different perspective.
One thing from my reading on the history of forms, is that the Japanese teachers quite often did not teach these applications to their Korean students, due to a combination of language barrier and racism. Koreans learned Karate while they were busy being occupied.
Japanese teachers quite often did not sponsor Koreans to open schools in Japan, teaching their own style to other Japanese. Something was different about General Choi and his relationship with Funakoshi, for him to be allowed to do this. While many Koreans did learn Karate while being occupied, General Choi went to Japan, and learned there... and then taught Shotokan Karate to other Japanese. I don't believe that Choi and the other TKD founders were as clueless about the Karate they learned as you seem to believe they were. I believe that they had quite a good understanding of what they had learned... which explains why Funakoshi supported Choi in opening a Shotokan dojo, in Japan, to teach other Japanese Shotokan. It also explains why they could iterate on Karate and the other arts they learned, both Chinese and native Korean, to produce such a successful art as TKD. These guys truly understood what they had learned and what they had been taught.