Ideally you want to root somewhere. Even with rootless chord voicings it's generally assumed somewhere in an ensemble the root will be played.
As you can see, the Em7/A has no E. Is this okay because the previous chord is Em7 and does contains the note E?
Yes, this is what I hear, a continuation of the Em7, because all the right hand tones are held through the second measure.
But, if you are relying on that effect, and if you already have notation, what is the purpose of the jazz chord symbols?
That isn't a rhetorical question. From the chord symbols about the only thing you might count on is the bass tones will be E then A. What happens above is a mixed bag. If a guitarist or pianist play from those symbols you could get very different results. Is that OK?
I imagine an important question is: do you specifically not want a full close voiced Em7 in both bars? If so, G/E and G/A closer to what you want played, even if it isn't necessarily the right harmonic analysis.
When I see piano score with jazz chord symbols, or guitar chord diagrams, above staff, I think Hal Leonard arrangements of pop/rock songs. Usually, I find those arrangements unsatisfying, often loosing the importance of the original arrangements. Are arrangement and voicing particulars essential for your music?
I think the question isn't whether it's necessary to include the root in a slash chord, but whether something like "omit the root in RH" or "play the same voicing for two measures" is what you really want.