I think this is just a mechanical thing. It becomes clearer when you continue your point about an A form chord.
Start with and open E chord, transpose it up by fourths, add fingerings to include all six strings, note the available basses and inversions...
E 022100 root pos only A 002220 2nd inv & root pos D 200232 1st inv, 2nd inv & root pos G 320033 root pos only C 332013 2nd inv & root pos F 133211 root pos only (start of barre chords) ...
(If you do this same kind of transposing by fourth with a fully closed position chord, like a C as 332010, the problem is even worse. The next transposing up would be F as 533211!)
Notice the problem when you get to C, you need to fret 5 strings! It isn't practical.
When you get to barre chords the E and A forms are practical, G is do-able with a big hand or high up the neck, but D and C form aren't practical. Those forms - E, A, G - allow for only root position or 2nd inversion. With no practical way to use all inversions, and with 2nd inversion being a harmonic special case, there doesn't seem to be a reason to go beyond root position in a chord chart.
Of course you can play inverted chord, but you need to skip strings, like x5x36x, and you would play with fingers instead of a pick.