Just because the guitar has six strings, it doesn't mean you have to play all of 'em. The essential chord is a "triad", three notes including the root, third, and fifth note of the respective scale.
If you limit yourself to one finger, your'e reduced to "barring" the three notes which greatly limits the number of chords you can use.
The 2nd-fret "A" major is a good example of a three-note shape that can be translated up the fingerboard.
However, it would be very difficult to do a wide variety of chords without adding other fingers.
Yeah, there are a lot of things to be played on the guitar without using all six strings. You can get many three note chords on adjacent strings and otherwise (5th-3rd-1st, 1st-2nd-3rd, etc.). Various tunes various various sounds and note combinations, so it all depends on the style/genre you are playing. As mentioned, sometimes in jazz you need to play more sparsely, for facility and to leave room for the piano player. In fingerstyle, we leave out strings all the time, in rock a tune will sometimes use power chords only and so on.
What people commonly know as "THE" G chord, is just one of many G chords. It is known as the "main" one because it is in open/first position and has been a starting in many methods and ways of thinking/teaching. Essentially, a "chord" is the sounding of two or more notes at a time. Personally, when I started sitting down a trying to really know/memorize the names of all the voicings/chords I like to play, it helped a lot. In other words, yeah its a good idea to learn many 3-note chords as well, also in that it puts off for a while the learning of the bar chord for the beginner - a good stress relief at the start of things.