Wow, where to start? Well first, let me say that this is a very complex song from a harmonic standpoint. That being said, don’t look for a heck of a lot of long term tonality or traditional harmonic function in this or any later Steely Dan song. Just satisfy yourself with the interesting and unusual chord movements and voice leading produced by those great chord progressions. However, if you look you will find many snippets of harmonic function in these types of songs
Regarding the chart, it is absurd to put this in 6 flats. This song is so chromatic that it makes no sense to use a key signature. This creates enharmonic horror in places like the E5 (Ab, Cb, Eb ?) and A7#11 (Db ?) bars to name a few. It also makes it harder to read, keeping track of all those flats when so many bars have naturals or sharps. It should be in an open key signature, all accidentals added as needed.
Now, getting to your question, the E triads sound like they might be G#m triads with a B on top to me. There is only one note difference between the two but then it would contain the maj7. F#/E gives you an Emaj7#11 flavor, especially when prefaced by the E (or G#m) triad. That enforces thinking of these bars as Emaj7#11.
Let’s skip the A7 and go to the Eb chords. These chords are really Ebm11 chords in my book. The Gb’s on beats 1 and 4 sound like a part of the entire 2 bars. Even though those Gb notes do not sound for the entire two bars, they linger in your ear until the chord changes.
That gives us Emaj7#11 to Ebm11. This is a bII to Im. The bII is a Neopolitan chord that is often used in Flamenco music and many pop tunes.
Now for the A7#11, it is not a substitute dominant because then the dominant chord would be Eb which is our target chord. This seems like a chord that has no real function, maybe they just liked the movement and the sound. I do see a micro function in there though. If you tonicize the E chord, then it is a IV7 chord. It doesn’t resolve but does have interesting movement to the next chord. The IV7 is widely used in jazz, pop and blues.
These chords are actually similar to the chorus of the song, except that the chorus starts on the Im instead of the bII and is a half step down, Dm11-Ebmaj7#11. They also don’t use the passing Ab7#11 chord on the chorus.
Some might not agree with my take but analysis is after the fact and can be somewhat subjective at times. Also, I’m pretty sure Donald Fagen and Walter Becker were not sitting around writing Roman numerals on a piece of paper when they wrote this :)