At what stage do we need to use other chords, like ii, iii, vi, or
To cut through the academia: we need to use other chords when they make the sound we want to hear.
Let's say you're singing in the key of C major.
Sing a G over a C chord. It sounds a certain way.
Sing a G over a G chord. It sounds different.
Sing a G over an Em chord. It sounds different again.
Sing a G over an Am7 chord. It sounds different again.
Those are all chords in the key, which contain a G.
Which of those is correct at a given moment in the melody? That's entirely subjective. You can change how a melody is perceived by changing the chords below it.
Of course, familiar tunes usually have a familiar chord progression, and if you deviate from that, it will jar to the audience.