As a mediant chord the iii chord has scale degrees ^3 ^5 and ^7. If we add the 7th (^2) it will have all degrees of the scale the serve a dominant function. Can it be said then that iii7 is actually a dominant chord? ALready, the theory textbooks say that iii6 can readily serve as a dominant chord without the ^2 because the the 6th of the 63 chord is hardly enough to kill its dominant function. However this is when the iii is in inversion with the bass note being ^5. This is not what I am asking. If you break apart a V chord you have 3 scale degrees which all pull to the tonic. ^5 points to the tonic by harmonic descending 5th, while ^7 and ^2 are upper and lower neighbors to ^1 and are the melodic tones most active to ^1. With these 3 tones in the mediant 7 chord, even though the bass is ^3 does it invalidate the other 3 scale degrees that all move to the tonic or can the iii7 be said to have dominant function in root position as well?
To be a dominant chord, it needs to first be a major triad; with a seventh, it needs to me a major triad with a minor seventh.
If we assume you are in C major, the iii (E-G-B) is a minor triad. With a seventh, it is a minor seventh (E-G-B-D). Neither of these will sound like a dominant.
If you were to raise the G to a G-sharp it would sound like a dominant, but it would "point" towards A minor, the relative minor. We would call it the V/vi. This is a fairly common occurence.
If you were instead to lower the B to a B-flat, it would sound like a diminished triad. This would lead us to F, and be called the iii0/IV.
EDIT: After @Todd Wilcox's comment of
"Can you elaborate on why, in the key of C major, the iii7 chord doesn’t sound and function like G/E? Or perhaps G/E no longer functions as a dominant chord with the altered bass note?"
I don't know for sure, but this is where my brain goes. In C major, the pitch B, or scale-degree 7, sounds very unstable. It wants to resolve upwards to C. But the interval of the perfect fifth has a stabilizing effect. I think (and this is just a hunch; I don't know how I'd prove it) that the addition of E in the bass provides a sense of stability. E sits a perfect fifth below B and gives it some context. That doesn't mean that you couldn't us this chord in place of a dominant. It just would never be as strong as a true dominant.
One way to approach the analysis is how the chord fits into the progression. In major keys
iii often progress to
vi, roots by descending fifth, working in the secondary/modal chords region. Of course the strong progression from
V is to
I. So, to make
Vadd6 clearer, move to
I. If you want
iii6/5 clearer, move to
German theory does have a concept relating
iii to either the tonic or dominant chords. Dominantparallele or Relative of the dominant seems to be the one your aiming for. But I don't understand that theory in any depth.
After re-reading the question I think something needs to be added about the specific inversion. When doing analysis I think the simplest explanation is often the best. Along those lines, if you have tones that are arranged to match up with a root position chord, it's probably better to analyze it as such, instead of an inverted chord.
iii7 is root position. Analyzing it as a kind of
V means you need some inversion, but in order for the tone in the bass (the
^3) to be a tertian chord tone, it needs to be the thirteenth, or we need to elevate an added
add6 to chord tone status, and putting it in the bass is analogous to a seventh in the bass. Chords extended to the ninth and beyond, in common practice, are not inverted. Or,
Vadd6 with the sixth in the bass is like a
V4/2 third inversion.
iii7 is simpler than a supposed inverted thirteenth chord with the thirteenth in the bass, or a quasi third inversion chord.
If the tones were arranged as
Vadd6, for example
C major, with
G the dominant as supposed chord root, I think you have a better case for calling the chord a dominant with added sixth rather than a first inversion minor seventh chord on the mediant.