@MrFox, it’s OK to do this. Being able to mute using either the thumb or the 3rd finger may come in handy for a particular song, but I wouldn’t let this stop you from moving on in your journey.
I strive to mute the low E using both methods, but the thumb is my most consistent method right now.
I read through other comments, but I’m curious to know if anyone having trouble doing this has seen improvement. I can’t play a D/G without muting the high E. A chords are ok (although a lot more effort to play it this way), C chords I just mute with the third finger instinctually.
Is it possible that I’m just anatomically incapable of performing this feat? I definitely see the value in it so if I can make it happen, I want to. But I also don’t want to waste valuable practice time on something that will never happen for me. Any advice is appreciated. 
It’s interesting…I have he exact opposite challenge: I can play D/G just fine, but always mute the high E on A chords. (I can usually get Am, but not A.) It was suggested to me to be aware of where your thumb is coming behind the neck…mine emerged up around fret 1. Moving it down to fret 2 changes the orientation of the fleshy part of my hand (the part muting high E) with respect to the neck. Not sure if that will work for you (we’re all different!) but it’s worth a try. I still don’t have it nailed, but I have hope!
I’ll double-check where my thumb is lining up. Appreciate the thought.
I had this practice item in my daily routine for about a week and a half. I thought I tried everything. I eventually removed it because the objective was to “explore” it and I didn’t really know how to explore it more. Thanks!
Jeff
I hope you two mean F#/D. Thumb over the neck you play the F# note.The note G isn’t in the D chotd.
Or are you just muting the E with your thumb, if so the chord would be D not D/G
Yup, we (or at least I) do! Was thinking about that after I wrote the reply. Thx for looking out for us! 
Yup. That’s what I meant.
I hope you mean D/F# 
Oh. My, Goodness! This is as difficult for me as playing guitar for other people! 
Yes, we understand, and we really DO mean: D chord with an F# bass. Really!!! (Hopefully I got it right this time!)
Jeff, I suppose it is possible that you have an anatomical constraint. I’m not in that field so couldn’t say for sure. I can say that when I first tried D/F# (this is correct notation) it felt impossible. A few years later I happened to come back and try again and I could play it.
So I think sometimes, just move on. Not because it will never happen but rather to continue to learn and improve. And at some future point you may discover that it is possible, just as an unexpected side-effect of everything else.
Spot on, Judi, that’s just how you say D/F#.
My hands are average size but my thumbs can only bend at a 45 degree angle which makes it impossible to not mute the low E string while playing the A chord for example. It has always been that way. Not much of a problem in regular everyday life but for this specifically it’s a problem. I guess I would neeed surgery of some kind.
Did you mean to mute the low E?
Well, you are supposed to mute the low E when playing the A chord, so it sounds like your thumb injury is actually a benefit in this case. 