Kostka/Payne is one source that explains the
iii chord is used relatively infrequently. But they don't simply say it isn't used.
You can account for
iii's use (or any other diatonic chord) through various harmonic sequences.
The "falling thirds" harmonic sequence (a.k.a. Pachelbel's Canon) is one good example of
iii in classical music:
[I V][vi iii]...
Another sequence using
[I V][ii V/ii][iii V/iii]
And of course the most well known, circle of fifths:
[I IV][viio iii][vi ii][V I]
So, you might ask why there is even this notion that
iii is relatively infrequent.
I offer this quick overview from the classical style:
The primary target chords are
I. They are the two tonal pillars.
The most common root progressions are: descending 5th, descending 4th, and ascending 2nd. Those progression leading to
I give use the following...
ii (desc. 5th) V
I (desc. 4th) V
IV (asc. 2nd) V
V (desc. 5th) I
viio (asc. 2nd) I
...I'm glossing over
IV I but that isn't really important for the conclusion.
The progressions above give us the subdominant chords
IV and the dominant chords
viio and the tonic
vi are left out from that basic list of progressions. You can call those two chords the modal chords. Too much emphasis on them would shift the tonality away from
I and into a potentially modal harmony. That sound is actually prized in pop and rock music, but it isn't the mainstay of classical style.