I want to start my answer by saying: I don't know. This is a very profound question. It does not have a trivial answer. I don't think anyone knows the answer for certain.
Because your question isn't just about how Common-Practice Era Western Music Theory Type analysis proceeds. It's about how we think about what a chord is. What is it about how we hear chords that makes us say that CEG is in an important sense the "same thing" as EGC? Why do we conceptualize the various inversions of a chord as, er, "a chord"?
So this is a hypothesis: we think about music as we hear it, and individual pitches rest on a fundamental pitch because of how physics work, so perhaps we bring the same thinking to harmony. I hypothesize our thinking about harmony proceeds from the assumption that harmony is really multiple voices making a single pitch with complicated overtones.
When you hear "EGC" (the first inversion of C major), even if produced by three separate instruments, maybe you hear it as if it were a single voice singing a single pitch with those overtones -- but we know intuitively that in nature that cannot be: no single resonant pitch can have the structure of EGC, because overtones simply don't happen in that order, so it implies to us an unheard lower C supporting that chord.
That's my guess.