Thus, when I hear things like "Of course, a rootless 3-7-9-5 so-called I7 (E7) chord is actually a diminished seventh chord (minor third, diminished fifth, diminished seventh (major sixth)) viio7 (D diminished 7th chord)" it makes sense in my mind what you're saying, but I'm also thinking... if someone showed you how to play it, rather than attempted to 'teach' you by showering you in theory, there wouldn't be a whole lot of confusion.
At the end of the day, the American styles of music are ultimately all aural traditions, whereas 'classical' has -- unfortunately, in my opinion -- become transformed into a written one.
But going back to the E7. Only the piano pays Gsharp, B, D, F#/F. The Bass player will play E, so while it's rootless on piano, it's not rootless in a band. Which is probably another reason you're confused. Jazz comping should not be played without a band. If you're just playing jazz piano then you should certainly play the root at some point (unless you want to sound cerebral, in which case don't do that).
> My advise to people who are learning music theory is to ignore terminology as much as they can and focus on intervals. Intervals are the true source of truth of harmony and everything else is someone's opinion as to how a particular structure should be named, perceived or modeled.
I always tell people attempting to learn improv and piano to 'play what sounds good and don't play what doesn't.