In a book that spoke about getting comfortable with all sorts of tonality and sounds, the author gave the example of an exercise to keep the triad constant while varying the bass note through all 12 note names. (The book also gave the exercise of keeping the bass constant, yet varying the major triad above through all 12 to see what would result.)
I specifically used "triad" in the first paragraph, yet many times in playing situations we use the word "chord" for any kind of multiple-note grouping including 3 and only 3. So next . . .
With your E chord, the author's example would be to see and hear what chord is produced by varying the bass under E. We see it by writing each on staff paper. We hear it by playing each. Going through all 12 it would be E, E/F, E/F#, E/G, E/G#, E/A, E/Bb, E/B, E/C, E/C#, E/D and E/D#.
Some of these will be very dissonant, yet may produce the desired effect within a given song.
Some of these may already have different names. For instance, E/C# is C#m7.
Some of these will be good to use in places where you need to connect one chord with another. Alone they may not sound intriguing, yet in some places the E/G# would make more sense to lead from or to than hopping to straight E.
Some of these will be the chord of the moment. In some musical times like the 70's, the E/F# type of chord was very popular.
In this set, my personal favorite is the E/A chord. You spoke about 3rds and major vs. minor. What I like about E/A (and the same chord type in other keys) is that it specifically has no 3rd laid out this way. Adding the 3rd, however, would change its characteristic. Would the 3rd be C for Ammaj9? Or C# for Amaj9? I want neither, I distinctly want the E chord sound above and the A sound in the bass. It provides a certain kind of tension, just like your E/G does too.
In other eras, it was forbidden to mix certain notes. Religious institutions prohibited it. In later times you can mix any notes you want.
Like that book I was talking about emphasizes, your E/G is fine for certain kinds of music. I seem to recall the band Rush uses many of these slash chords. A lot of times the whole slash thing is to emphasize something different is happening in the bass than we usually expect from the rest of the music. Music theory serves us well to understand certain concepts, and then we have the freedom to break it.
And sometimes, if you want to leave your audience hanging, purposely end with one of these chords where the bass doesn't "match." Finish your entire song that was in E with your signature E/G, either quickly or with a fermata and say nothing after that.