"The game" can mean too many different things. In one place, I might (mildly) insist that "the game" does mean "just the mechanics", as when I say "I'm not sure if I still want to play the same game, or if I'd prefer to play Shadowrun instead". In another, I might insist that it has absolutely nothing whatsover to do with the mechanics!
This is one of the reasons I talk about how games (of all stripes) are designed, and thus have elements of technology and technique, in addition to elements of aesthetic or taste; that design is pointed at trying to invite or foster a certain experience in those who play them; that a game design goal is distinct from the inspiration which leads a designer to begin designing; and that there can be distinct game-design-purposes toward which specifically RPGs usually tend. It's also why I adamantly insist that it is unwise to conflate (what I call) "emulation" with (what I and many others) call "simulation", for example.
If we're going to really, deeply talk about a particular TTRPG, as opposed to just lightly dancing around its contours, we're probably going to need to be more specific than just throwing around casual terms like "the game."
The experience of any game is always more than the rules of that game, because rules on a page are dead things. They require human effort to have motion, life. It is not demeaning to describe, in clear and unambiguous terms, the tools that a TTRPG uses in attempting to cultivate a particular experience in its players. Naturally, the tools that attempt to cultivate that experience are distinct from the experience itself, just as my skillet is distinct from the flavor of the food I cook with it.
Many games that (often proudly) call themselves "old-school" are of consciously minimalistic bent. Why, then, should it be insulting to note that their core, fundamental process is, itself, minimalistic? A minimalist aesthetic can be breathtakingly beautiful in the right context. We do not assert that architecture is inherently insulting for describing minimalism as...intentionally not using many elements. (As a good example, I will never like Brutalism as we actually have it--but in its highest ideals, it can actually be quite beautiful, all the more tragedy then that its boosters almost always neglect all the things that were required for those highest ideals to manifest in actual buildings!)
Or, if you prefer: The highest and most succinct tier of describing the art of swordfighting, which has more variations and beauty than I could ever know, is "by blocking and chopping, thrusting, or slicing, kill the other person first." Undoubtedly, thousands of styles, perhaps millions of maneuvers, literally thousands of years of technique and style, boiled down to a relatively pithy sentence. I suspect others, who know the art better than I, could trim it further.
To respond to that with, "But you agree that the campaign isn't just the bladework?" is...well, it's injecting an entirely different conversation in, because it (seemingly?) upsets you that someone focused on an analysis of combat isn't talking about the entirety of war every time they mention to any degree what swords do.