Actually, the fifth is the least essential component of a dominant seventh chord; the (dissonant) tritone interval between the third and the seventh is the most characteristic and vital one—to the point that you can use just these two notes to suggest the entire chord.
After the fifth, the next best choice for omitting would be the root (not the third), leaving just the third, fifth and seventh. You'll often find fifth-less and rootless forms of dominant seventh chords in fingerstyle and chord/melody arrangements.
If you want movable shapes for C7 which include all four components, you can use 3433 or 5767 (or even 5431—yes, I do occasionally use this shape, particularly in linear tunings).
In re-entrant tuning, a simpler way to play C7 which sounds the same as 5463 is 3463 (where the index can barre to stop both 3 positions). But in linear tuning, 5463 or the full forms are generally preferable: 3463 doubles the 7th at the octave, and puts one at the bottom of the chord, which increases the sense of instability. 5463 doubles the root, with one root on the bottom, the most stable voicing.
I assume that the spelling mistake (A# for Bb) was imposed by the program being used to generate the diagram. It probably represents all the accidental notes by their sharp variants, ignoring the third stacking behind the construction of most chords. Your pointing this out isn't "super-picky" at all: it's things like this that make me hate most music helper apps—the programmers don't sufficiently understand music, so they lead others to replicate their quite basic errors.