In my reading, a criticism that I have heard leveled against Neo-Riemannian theory is that it does not explain the smooth chromatic voice-leading possible in the iv - I instance of the plagal cadence, often played IV - iv - I.
This reminded me the famous/infamous "Creep" progression, I - III - IV - iv. Imo, there is a interesting duality in the validity of analysis from a functional perspective, but also from a inter-relational (with respect to chords) perspective. One could analyze it at the movement from tonic function chords, I and III (debatable), to subdominant function chords, IV and iv, and back again. Conversely, from the perspective of chord proximity, each chord maximum distance from the next is two semitones. Further, if you minimize the movement of individual voices in the progression you can see three lines moving a half step at a time; 1 - 7 - 1 - 1, 3 - 3 - 4 - 4, and 5 - ♯5 - 6 - ♭6.
Is there a school of theory that deals with harmonic proximity in a tonal context? Or how functional harmony relates to chromatic motion? Or in general the movement of important voices to connect chords?
Also, is my analysis of this progression correct? Does any form of analysis generally considered to take precedence?
Edit: As @Dekkadeci points out, there are other ways to write the progression down using roman numeral analysis. He points out the I - V/vi - ♭VI/vi (IV) - iv interpretation. Further, within my original analysis of the chord progression. I have heard two different explanations for what the III does. One is that it is borrowed from the relative harmonic minor. The other is that it is just a III chord. Personally, I think it's just a III chord.