1st inversion of C is E G C. Is that not a type of E chord? Why are inverted chords not named by their root (1st) note?
To directly answer the question - why isn't EGC (1st inv. C major) some sort of E chord?
It could be. E and G gives an E minor feel to it. Add C calling it B♯, and the chord would be called Em+. A very rare chord, but technically that's its name. Putting them in 2nd inv. (GCE) trying to make a G chord? The C would be a suspension of B, but the E isn't anything suspended - it's simply an added M6. So, at a (big) push, it could be called G6sus4 - a chord I've never come across. Mainly as with those 3 notes, it's universally recognised as 'ah, it's C major!'
Theory is there to try to explain what's happened - so that in the future, it makes it more plausible, more understandable, for future users.
The blend of ^1 ^3 and ^5 is so ubiquitous now, that we all accept the fact that whichever order those notes are stacked in, they constitute a simple C major triad.
All that apart, as Aaron says, the triad comprises R, 3 and 5, which translates (in key C )CEG. Play or write them in any other order, and while the lowest note may differ, the R 3 5 pattern will only work from the 'real' root of the chord, here, C.
This is a fundamental music theory concept: (root position) chords are defined as "stacks of thirds", and inversions are considered to be "the same chord". The notes C-E-G are C major regardless the arrangement in which they appear (E-G-C is not a "stack of thirds"; it's a third and a fourth, so by definition E can't be the root).
The underlying justification for the theory describing C-E-G and E-G-C as both being C major is that composers used those chords in similar contexts and in similar ways. As just one example, one set of pieces might have C-E-G moving to B-D-G while another set has E-G-C moving to D-G-B. From there, a theorist might observe that in both cases C moves to B, E moves to D, and G stays as G. That suggests that C-E-G and E-G-C operate in much the same way and thus serve the same (or nearly the same) musical purpose. Further observations about how the notes C, E, G operate when used together leads to the theory that C-E-G and E-G-C are both C major.
However, where the original question is particularly interesting is with second inversion triads. G-C-E is sometimes interpreted as a C major chord and sometimes as a G chord with suspensions. As a starting point for investigating this issue, search this site for "64 chord".
The reason for this originates from the same reason chords, in the modern sense, exist in the first place.
For inversions to me named by the root, there has to be an assumption that the bass note is what primarily identifies the chord. This is rarely the case. Chords did not develop from "This is the lowest note, therefore this is the chord" and then have the two thirds following along become a norm.
Chords were developed because certain pitches mathematically "belong together" and are pleasing to the ear when played harmonically.
This is due to the overtone series.
When any note is played, what we primarily hear as one note is actually a collection of multiple pitches, with the pitch we recognize as the note, C4 for example, being the most prevalent. It is also, incedentally, the lowest note and referred go as the Fundamental pitch.
Why? Because of physics. Think of a guitar string. You can play harmonics on it, which are actually the overtone series of whatever pitch the string is tuned to. And, different harmonics sound more loudly than others. This is because they are the closest to the original pitch. The first overtone is the octave. With C2, as in the example below, that would be C3. The second is G3, then, C4, then E4,and so on.
As you can see, the first few notes build the C chord. There are more, but all together is a bit complex.
The progression of overtones was simplified, and around the advent of the modern keyboard and then sheetmusic as we know it, became notated as the tonic, third, and fifth for general use.
Suggesting that an inversion should be named for the bass pitch, then, completely overlooks why chords even exist. In E-G-C, G and C simply aren't in the overtone series of E, at least, not early enough to be audible or relevant to the pitch of E.
All three notes are, however, among the first registers of a C.
Other versions of chords, namely minor, diminished and augmented chords, exist for other reasons but are grounded in the structure formed from the overtone series.
This reasoning is often overlooked in modern music due to Equal Temperment, where the notes we play are actually out of tune from what they are supposed to be. The reason for this is that if instruments were to be in perfect tune, they could only play in one key at a time. As that would prove inconvenient, Equal Tempermant was developed, where notes are slighlty out of time to accommodate their existence in multiple keys.
It is actually a result of the oft overlooked overtone series that keys, the circle of fifths, consonance, dissonance, and most everything we know about music exists.
For more in depth explanation, and also sources lol, is this article below. https://intmus.github.io/inttheory18-19/08-overtones/a1-overtones.html
If you play a chord progression over and over, e.g. C -> G -> Am -> F, you could replace C major with E-G-C or G major with D-G-B, it will not change the song structure at all, and the original chords will still be clearly heard.
In a band
Inversions are also relevant when playing in a band.
As an example: if the guitarist plays E-G-C, either as a chord or an arpeggio, while a lower C is played by either the bassist or the pianist, it will sound pretty clearly like a C major chord.