The harmonic function of a broken chord is EXACTLY the same harmonic function as a block chord. There is zero difference.
The melodic function of a broken chord is to provide melodic interest. Block chords have almost zero melodic interest.
The rhythmic function of a broken chord is to provide rhythmic interest. Block chords have almost zero rhythmic interest.
There really is no rhyme or reason in music. It's done because it "feels good". It's like wondering why people have sex. Because they do. If they did not want to they wouldn't. People play broken chords because they can.
Much of music is that way. There are no universal laws saying that you have to do X Y or Z in music. You do things because you can and want to.
Obviously on monophonic instruments you only have the choice of broken chords and hence it is obvious they would show up.
Broken chords are also called arpeggio's. Again, they do more than just block chords(which only provide harmonic interest/color). All music uses them because it's just sounds and all music uses sounds.
In music, the primary goal is to express harmonic clarity. Chords are precisely what do this. Arpeggios do this but sort of stretch it out in time. Generally speaking a piece that is only of block chords is quite boring. 99.99% of all music consists of combinations of elements to make it interesting.
Also music is independent of instrument. The guitar is not special.