Unpopular opinion: if it walks like a duck and quacks like a duck, then dude, it's a duck.
Inverted chords have, in my ears, never sounded anything like the chords they are supposedly an inversion of. It's like saying "10 is just really 3+7". It's true, and it's also useless. Let 10 be 10. Let 3 be 3. Etc.
Most composers don't even use this system as they compose, it's just for after the fact analysis. Composers hear a tune in their head and then transcribe it. Unless they're writing cookie cutter shit without inspiration, then they might synthesize some soulless crap that intentionally follows some well known chord progression as picked by a dart thrown at a poster of the circle of fifths.
E-G is a 3 semitone interval and G-C is 5 semitones so it's an E+3+5 chord. If you wanna play it on piano, that means leave a gap of 2 keys and then a gap of 4 keys.
Known in western classical harmonization theory as "Em+" (E chord with minor third, augmented 5th). C first inv is actually Gsus4add6no5 or as I like to call it G+5+4. It has an incredibly warm and rich tone, nothing like C+4+3 (C major triad).
It works in C because all of its notes are also present in the C major scale, but that's just a coincidence. Calling it a C inverted chord is like making astrological or numerological conclusions.
And no, shifting up or down by octaves is not the same sound. It's close but not equal. You can probably get away with octave substitutions in a busy song but you can not say it sounds exactly the same. A 9th is not a 2nd. 11 not 4, 13 not 6. When you play La Campanella, is that your excuse for skipping those octave jumps? :)
It would make jazz lead sheets impossible to read and play. The whole point of a jazz lead sheet is that a chord symbol can be played in a huge variety of different ways depending on how the jam is going. If you followed the chord symbols literally and only played them all in root closed position then depending on the stage you might get a cymbal thrown at you.
Inverted chords have, in my ears, never sounded anything like the chords they are supposedly an inversion of. It's like saying "10 is just really 3+7". It's true, and it's also useless. Let 10 be 10. Let 3 be 3. Etc.
Most composers don't even use this system as they compose, it's just for after the fact analysis. Composers hear a tune in their head and then transcribe it. Unless they're writing cookie cutter shit without inspiration, then they might synthesize some soulless crap that intentionally follows some well known chord progression as picked by a dart thrown at a poster of the circle of fifths.