Why don't we see, for example, a clef with a stylised symbol representing an A, B, D or an E? What's so special about G, F and C? Who decided to use them and why?
Three clef signs are in common use, plus the neutral or percussion clef.
The most common are those found in piano music, where the two clefs are separated by a space where 'middle C' lives, on its own leger line when it's needed.Go up a 5th from that C, which has become a sort of data point, and there's the G line, around which curls the G clef - often known as the treble clef.
Go down a 5th from C, and there's the F line, around which curls the F clef - often known as the bass clef. I guess those notes were considered pivotal, and they're also symmetrically placed.
A lot of instruments can have all, or most of their notes written within one of these two staves, with not too many dots being on leger lines. However, some instruments' ranges fall awkwardly, so the C clef is used. It's moveable, and can make any line into the place where middle C lives. Don't usually see it depicting middle C on a space.
Other clefs exist, for example a treble clef with an 8 attached, which gives a useful place for music for guitar. All the notes are actually written an octave above their sound on guitar, which saves writing it out in bass clef, and then having to utilise leger lines.
The reason there aren't A clefs, etc., is probably that the C clef, being moveable, will do pretty well any job that a (new sign) A clef might, albeit one note name out. So we already have enough tools for the job. And loads of musos are happy using them; they don't confuse; music is working well with them.
The neutral clef by the way, is for mostly untuned percussion, where, for instance, a snare drum is given the middle line for its dots.
EDIT: according to 'The Origin of Musical Clefs', the treble clef started life as a 'G', the bass as an 'F' - quite easy to see - and the C clef as a 'C', although why isn't stated. C clef's fairly obvious, but as to why treble needed to look like G, and bass F, I haven't yet discovered.
Almost any convention could be different and, in many cases, probably better as well. However, in most cases, one has won and it is too disruptive to change.
Of these clefs, probably the C clef is easiest to understand even though it is the least commonly used. It marks the position of middle C which has become a standard reference point. This clef is used where possible. It becomes unavailable when you go as far up as the treble clef or as far down as the bass clef as neither of these has a line for middle C. I guess that a variant of the C clef could have been used to mark the C an octave above (treble clef) or below (bass clef) middle C but these would need to mark gaps between the lines which might be less clear. The next most natural interval is a fifth and this has the advantage that a fifth above or below middle C is also on a line. So, marking a fifth above middle C (G clef) and a fifth below (F clef) is quite a good solution.
Edit: Some care is required with the word "clef". It is used in two slightly different senses: referring to the symbols (as you are using it); referring to a particular symbol in a particular location. For example, the "alto clef" and the "tenor clef" both use the C clef symbol but in a different location. Today, this is only common with the C clef symbol but it can occur with the G and F symbols. This article seems quite good: Clef (Wikipedia)
Edit: I forgot the percussion clef, see Tim's answer.
When first trying to show absolute pitch levels, European musicians did use a single line through some note-predecessor marks to represent a particular pitch. The earliest "clef" marked the mi-fa semitone (the only one if one considers octaves to start over at C rather than being connected to B below.) Other lines (and clefs) followed. The signs used (F,C,G) for clefs are basically a historical accident (in the sense of a random happening) though marking the semitone did yield a rationale. One could guess that were musical words written using a futhark, the clef signs may have looked different.
Clef does come from the root for "key" (the lock turning device, not the islands) as in the "key" to a puzzle.
What's special about G, F, and C, is that they are the keys in the middle of the system: C has no flats or sharps, F has one flat, G has one sharp. As such, they were the most natural notes to indicate.
Probably this is some sort of misunderstanding: While clefs were named according to their appearance, today this is far from obious. The note they are named after is of no special importance to that clef.
It is merely an indicator, where to put a characteristic feature of the clef (in case of G and F clef the inner center of the spiral, for the C clef its "nose"). It turned out, that the name is succinct enough to refer to the clef shape and so it stuck. Note, that only C clef maintained a variable position until today, therefore additional names were required like tenor clef (a C clef sitting on the second line counting from up to down).
The clefs, and their corresponding pitch location placements, have their historical origins connected to the ranges of voices and instruments, and the related problems of notating musical sounds. In particular, starting with the ranges of "typical" human voices, the frame of reference was based on the ranges (of basses, tenors, altos, and sopranos) that were considered "normal" for western art music and its predecessors. The point of the clef's placement is to provide a shifting window that ensures that the "bulk" of the notes (pitches) for a given piece of music and part (bass, tenor, etc.) end up on the staff, and not above or below it on ledger lines. The reasoning being that the pitches on the staff are easier to read and locate than those that end up on ledger lines above or below. It's as simple as that, really just a mechanical artifact of the notational system that was imposed.
Consider that if there was just one staff for all musicians and instruments, with all pitches on it absolutely positioned, that would take an awful lot of paper to write out the parts, and the bulk of that paper (vertically) would be "noteless" wasted space, with nothing but empty staff lines and spaces, no notes. The clefs (and the closely related situation of instruments being pitched in specific "keys") are really just a mechanism to solve this problem of notation. Not that we need to make it a contest, but with the exception of the C clefs being used for some relatively limited situations, one could say that the system of instruments being pitched in certain keys was a much more "successful" attempted solution since it persists to this day, while the use of clefs other than treble or bass has fallen off, noting that the overwhelming bulk of all music today is notated on either treble or bass clef (or the combo of the two).
There are lots of examples of instruments that, over the years, have moved around from staff to staff. Johnny Smith had a guitar method in the 50s and 60s that had guitarists reading off of the grand staff (same as piano) while current guitarists read treble clef although the instrument is transposing down 1 octave from where it is written, similar to the tenor sax, which is transposing down a major ninth (thus Bb tenor sax). Baritone horn (Euphonium) and tenor trombone can both have parts written either in treble clef Bb and transpose down a major ninth or they can read bass clef at actual pitch (in C), and if you play those instruments, you are prepared to deal with either.
Bach, the famous composer, originated our present form of notation. There is the treble clef (the "G" clef you mention), the base clef ("F" clef), and the movable clef, which can be tenor, or alto, or anything. Up to recent times, there were more clefs. For singers, there were four different ranges; soprano, mezzo, tenor, and bass. The "Grand" or great staff, with eleven lines, contains both the G-clef and the F-clef.
The reason they have survived to the present day, is that they serve the scales of most instruments. Even so, there are many instruments that use the movable, "C-clef." The "tenor" clef is used for the upper ranges of the bassoon, cello, euphonium, double bass, and trombone. The viola uses the "alto" clef. There is even a "sub-base" clef.
For a history of music, youtube currently is hosting 'Howard Goodall's Story of Music'. It's a six part documentary, that goes from Roman times, showing the origins of notes and scales, up to modern day.
Also, for a classic on notation and ear training, George A. Wedge's book - 'Ear-training and sight-singing applied to elementary musical theory, a practical and coördinated course for schools and private study' is free online, and very inexpensive on Amazon. Go to https://archive.org/details/eartrainingsight00wedgiala for a preview of it.
Hope this helped. Good luck with everything. B'H.