The open C Major Chord is one of the first basic chords taught to beginning guitar students. It is also one of the most common chords I see in the music I listen to.
The instruction books teaching the fingering for the open C Major Chord stress the importance of avoiding the open low E string when playing the C Major Chord and the chord diagrams for the C Major Chord always show the 6th string being muted.
But the E is the third of the C Major Triad (part of the chord) and playing the low E string on the open C chord would be much easier than trying to avoid it with your pick or mute it with a finger tip. I realize that if you included the Low open E string in the chord it would be a first inversion of the C chord or could be referred to as a C with E as the bass note (C/E). But it is still a C Major Chord (just a different voicing) as explained Here and Here by two of the most venerable contributors to this site.
So why does it seem that guitar instruction books make it sound so wrong to play all 6 strings when playing the open C Major Chord?
And why is it rarely used in composing music for guitar?
I often see the second inversion of the C Major Chord in music composed for guitar (C/G) but don't remember ever seeing any song I have ever looked up the chords for that called for a C with E in the bass (first inversion).
Many of the songs I have learned to play use the first inversion of the D Major Chord D/F# and I play that chord in many of the songs I cover and it sounds good.
So why is the first inversion of the D Major Chord far more common than the rarely used first inversion of the C Major Chord which is certainly easier to play than the D/F sharp because it's just a matter of strumming all 6 strings?
Is it just a coincidence that the songs I choose to look up the chords for just don't have the C/E chord as part of the arrangement whereas many of the songs I look up do specify the D/F# chord?
Or is there something about a C/E that diminishes its use in music written for guitar?