So the index finger is supposed to hold down all the strings on fret 1, but mute the low E string, but isn't fret 1 of low E an F note? Why not just play the low E string's F note as well?
I'm not quite sure where you got your idea of an F barre chord, but both are correct. The F barre chord tends to be the hardest E shaped barre chord for beginners who don't have their finger strength up, so there are many varieties that do not have all six strings playing at the same time. There are also other flavors of Barre chords that will give you different voicings, but I'm not going to go to get to that in this post.
Taking a step back from the exact shape of the bar chord, you only need three notes to make an F major chord which are F, A, and C. If you look at the full E shaped F major chord, you can get away with only playing the high three strings as that has all the chord tones tones in it.
Never seen a barre F chord at fret 1 not play the bottom string as well - most will insist, as beginners are encouraged to play root position chords, and starting on the 5th string will make it a second inversion. Not wrong, but not sounding as strong.
You may be thinking about a Bb chord, which is often shown with a muted 6th string, although that again can be played , making a second inversion of Bb.
And - the first finger is not supposed to hold down all the strings - there's no point in it holding down a string that's fretted above the barred fret, so no point in pressurising it to do so.
The first F-chord I learned (self-taught) was the non-barre version which is basically the barred-F but you use the index finger to hold down the E and B, and then do not play the low note.
If this is the F form the question is about, then the answer is that this leaves only the thumb to fret the E.
I have big enough hands that I can do the "Hendrix Barre" where I fret the E with my thumb (if I choose to use this form). Many people do not, and so the solution is to not play the unfretted string (open E).
The image below shows the "F form" I refer to above, as well as a "Hendrix Barre" example. Hendrix appears to be fretting 3 strings with his thumb.
Is it possible that the chord diagrams you're implicitly referring to are only showing a single instance of each note of the chord.
In such an instance it would be serving to drive home that the (major) chord is specifically three notes despite that up to six notes can be strummed.
(Has anyone seen any chord box diagrams where perhaps the main notes are denoted in black and the optional ones in grey?)
There are no rule to mute it. If you want to get more bass - you can play it with barre, if you want more funky sound and more control - you can press separate notes. And lets clarify terms - "Chord" is not about "how to play some notes", its just a grouping of 3 or more notes that's sounds in tune in some scale. How to play them - it's about "fingering" of chord, and it totally depends of your tonal and comfort preferences.
So, F chord consists of F, A, C notes - play them in any way you want)