When I look at SpecKit, I see a kind of vibe coding fantasy: "code is no longer king", stop writing "undifferentiated code." There is no code on the site, just a bunch of prompts and commands.
On the other hand, what you are describing above is bringing specs closer to the codebase, while not replacing the code itself. Like I said I have no problems using natural language as a guide (even as a primary guide). I also completely agree that it helps with documentation.
My main point is: if you want to maintain a complex system, you also need to have an accurate description of the system behavior in some kind of formalism.
This kind of description reflects the true system behavior better. It's more helpful when you need to predict the impact of changes and also during debugging.