Question about Diatonic Chord Substitution

I learnt about Diatonic Chord Substitution recently, and I understand most of it. Taking the key of C, Am (vi, C's relative minor) can replace C because C E G and A C E are only a tone apart (A-G). I understand replacing the IV chord (F) with the ii chord (Dm) because F A C and D F A are again, only a tone appart (D-C). However, why can't this principle apply for the iii chord Em? Em is G's relative minor (E G B v. G B D), but instead only vii0 can substitute for the V chord. Is it because there's only a semitone of difference between the iii chord and the I chord instead of a whole tone, that we naturally substitute the iii chord for the I chord? Wouldn't it be able to substitute for both the Tonic and Dominant chords?

Thanks for the help (I'm sure it's a stupid question but it's been nagging me for a while now)!