I am usually doing some form of fingerpicking pattern (like Travis, Arpeggio or Carter Lick) with an alternating bass, so yes, as pointed out by several people, the bass E is not an appropriate note to start the A chord. But with that said, the thumb pick usually alternates between two strings, and the bass E is fine for the alternative thumb pick.

I brought this up because I noticed that, in chord books/sheets, the bass E is usually shown muted, even though the treble E is played open, and that seemed counter-intuitive to me.

If you use alternative bass picking and do NOT play the bass E in an A chord, you would most likely alternate between the fifth and fourth strings, and in an A chord, the fourth string would be playing (gasp!) an E - so it would seem to also make sense to start on 5 (A) and alternate to 6 (E). When I do this, it sounds just fine. So why do my chord diagrams seem to indicate that this is verboten?

Probably I'm just being picky (no pun intended), but isn't that - to some extent, at least - the privilege of the Mudcatter?