For example, in the key of G, why is the sharp not put next to every F note? I think this would make it easier to sight read quickly, especially for keys that have many sharps or flats. Is there an advantage to having it the way it is (indicated by the key signature)?
It is related to "chunking", once you are used to keys, it is easier to quickly understand the single chunk "This piece is in G major" instead of having to see and interpret each of the individual sharp signs. This aids sight reading.
With the way keys are conventionally notated, the presence of accidentals is actually informative: it tells you when the music is doing something outside of the home key. If every alteration were notated, you wouldn't get the same kind of clear visual indication that these passages are doing something outside the home key.
It wouldn't be easier to read.
Firstly, most instruments are not tied into any particular key signature.
A simple sequence like someone singing/playing a scale in E major and someone else singing a third above feels very natural but actually is a complex walk resulting in an unregular sequence of major and minor thirds.
This makes sense and can be done in confidence when performing it in the context of an E major scale.
In general, the decreased readability leads to more confusion and less complex. The problem is that the second version is not even a neutrally notated version expressing an E major melody: it is expressing E major in relation and contextualized to C major! This would cause issues to instruments that are naturally in other keys, such as b flat clarinets.
As a result, most musicians keep the key signature, and only augment the markings with their own dynamics and/or fingerings. Funnily enough, when playing this on a piano which has a rather direct relation of white keys to C major notes, even then this notation will grow old on you very fast.
And the reason is that even piano players will become at home with different scales and will be playing an E major scale without thinking like they do with a C major scale, and will think of modifications in terms of the E major scale.
For similar reasons you cannot make do just with sharps even on instrument with equal temperament but also need flats and double sharps and flats: so that stuff can be written in terms of a home scale and diversions from it.
More chromatic notations (they would offer themselves for things like chromatic button accordions) have not caught on because the major musical building block of our Western music are diatonic scales. If you are expressing the simple melodic phrase I spelled out in terms of their chromatic steps, it becomes rather irregular. The first third is major, the second one is minor, the third is minor, that of the first half note is major again.
If you express it in terms of the C major scale, things become even worse. The first third consists of the third and fifth of a C major chord, only that you raise the fifth by a semitone.
It's not just about which notes are flat or sharp
The key signature doesn't just tell you where the sharps and flats are. It tells you where the pitch centricities are, and tells you the function of each and every note within the scale. These are both important for interpretation, as the function can affect inflection, tuning, fingering/bowing/choice of string, etc.
Key signatures take less reading
A piece written with key signatures is going to have less ink on it than one where everything is treated as an accidental. If there is less ink, there is less to read, meaning you can read it faster. So key signatures actually make sight reading easier (assuming you know your scales).
OTOH, some composers do as you suggest
For some types of compositions (e.g. serial music, neoclassical, some forms of jazz), you may find that the composer has chosen to "keep it in the key of C" and write only accidentals, either because (a) the very concept of a key signature may not make sense in that particular work (e.g. it is atonal or does not use a diatonic scale), or (b) the key changes so rapidly that it is pretty ridiculous to try to write everything out the "right" way.
While a newcomer to the piano might feel like key signatures are essentially arbitrary, and may need to think about them while playing, people who are experienced at the piano get used to how different key signatures feel and don't have to think about what notes are sharp or flat--they just play them.
It may be helpful to think of sharps and flats not in terms of note names, but rather their positions on the keyboard. The first sharp is the leftmost note in a group of three black keys, and the second is the leftmost note in the group of two. Sharps alternate between the group of three and group of two, working toward the right. Flats also start on the group of three and alternate with the group of two, but they proceed right to left.
If one thinks of things in those terms, then what a pianist will be conscious of is not which notes have sharps or flats in the key signature, but rather how many notes from one side of each group of two/three need to use the black key that's toward the other side. In F major, for example (one flat), the note just to the right of the group of three moves left to a black key; none of the notes near the group of two move. In A major (three sharps), the white keys to the left of the first two black keys in the group of three, and the first black key in the group of two, move right.
There might be some instruments where individually marking sharps and flats for each note would be helpful, but for most instruments it would usually increase a performer's mental workload.
Helping hands in the long run make it more difficult to become proficient. As you advance in playing your instrument you will find yourself recognizing and playing patterns, rather than individual notes. Reading, interpreting and playing a "crutched" score can adversely affect your ability to play it in the long run. This is also true when helpers in the past have annotated the score with "helpful" fingering for notes.
Besides all the reasons above why key signature are better I'll add that using key signatures when you do see a accident it's like someone shooting up a warning flare that something is altered. Not only good to know for sightreading, but if comping it's a note to adjust for, same for improving a note to take into consideration.
It's letting you know a note needs special handling from multiple points of view.
You might want to consider joining the dark side: take a pencil and put small marks, for example a tick sign ✓, near the notes altered by the key.
I find that since a key always has either flats or sharps (or neither (or actually, it might have flats with double flats or sharps with double sharps, but what you gonna do...)), you don't need to care to draw the actual signs ♭♯, and a small mark or a scribble near the note will do just as well, and it doesn't clutter the score as much, since it's miniscule, and in pencil.
Be wary that you might not want to come back, so make your choice...
In many ways, music that adheres only loosely to the key does have the accidentals in front of the notes. Jazz music, for instance, does not stay strictly in the key as what is the norm with classical music. You can have a host of chords that have seventh that are completely foreign to the key but are just used to add colour to pieces. Sometimes these ideas are used so heavy in Jazz that they do look like they have all the accidentals in front of the notes.