We're dividing its path into small segments that the clock postulate lets us work with. Ultimately, that's why the interval is written using infinitesimals. We use them because in this form the interval embodies the clock postulate. The clock postulate is one of the ideas we use to build the theory of gravity known as general relativity. That's not true at all.
General relativity is built on a foundation that includes the clock postulate.
That would be circular reasoning. General relativity is built upon the ideas of special relativity, but the physics of accelerating clocks is completely handled by special relativity alone. No, the EP is fine. Again, the confusion here is the same sort of thing as above where we spoke about the wind chill factor. Let's try to see what is happening.