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.
I'm not sure what "explain" means here, but I'm assuming from the phrasing of the question that it is limiting the concept of "Neo-Riemannian theory" to the basic PLR operations. However, PLR are one possible starting point, based only on a mathematical theory and a few vague historical diagrams (among many other possible systems).
The assumption of the PLR system is that two common tones are held and only one voice is moved at a time. IV-iv is a P operation, as only one voice moves. However, to go from iv-I requires two voices to move, so obviously that motion cannot be explained within the simple PLR system. It's not a strong criticism as the PLR system doesn't incorporate such motions. It would be like criticizing the theory of triads because they don't "explain" seventh chords.
Nevertheless, there are plenty of options within the broad umbrella of Neo-Riemannian to expand the possible operations. iv-I is certainly rather smooth semitone voice-leading, so having a name for that operation might make sense if one is privileging parsimonious voice-leading. And indeed, in the Neo-Riemannian literature, that motion is generally given the name N after Nebenverwandt ("neighbor-related"). Riemann himself referred to this as a Seitenwechsel relation.
And N is certainly not the only chromatic semitone motion relation of that type omitted by PLR. Another is the S transformation (after slide), where one begins with the two outer notes (fifth interval) of a major or minor triad and move them both by semitone on the same direction. (For example, consider moving from C major to C♯ minor.)
One can create a series of consecutive PLR motions that are equivalent to two-voice motions like N and S; it all depends on what your goal is. For example the N motion from iv-I could also be analyzed as a combination like RLP or PLR. Saying that Neo-Riemannian theory doesn't "explain" this voice-leading in this case would mean that it also fails to "explain" voice-leading in things like the basic V-I cadence which also has a common tone and relatively smooth voice-leading but which can't be summarized directly by a single PLR relation.
This reminded me the famous/infamous "Creep" progression, I - III - IV - iv. [...] 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.
Yes, and these motions can be analyzed using Neo-Riemannian theory if one wishes. The I-III progression could be viewed as combination transformation LP, though some theorists have also proposed names for this two-voice semitone motion, most commonly M (for mediant). III-IV just slides all voices up by semitone, and there's not a very efficient LPR representation of that, but one could use SP. And obviously IV-iv is just a P.
Is there a school of theory that deals with harmonic proximity in a tonal context?
I'm not sure what is meant by "harmonic proximity." Given the question, I'm assuming this actually means something more like "melodic proximity" or "voice-leading proximity." But if "harmonic proximity" within tonality is meant, there are also plenty of models for that. Some attempts at empirical models like Fred Lehdahl's Tonal Pitch Space and some of David Huron's ideas about measuring harmonic proximity come to mind.
Or how functional harmony relates to chromatic motion? Or in general the movement of important voices to connect chords?
Especially since the growth of Neo-Riemannian theory beginning in the 1990s, there have been a lot of proposed models for thinking about voice-leading metrics, parsimonious voice-leading in progressions, etc. For a general start on classifying chromatic motion in functional harmony (with a Neo-Riemannian slant), one might start with David Kopp's Chromatic Transformations in Nineteenth-Century Music, which summarizes the history of harmonic models that led up to Neo-Riemannian theory, as well as elements like N and S and his own systems for classifying progressions according to voice-leading.
But one could also go in very different directions with these questions. By some standards, Schenkerian theory is all about voice-leading and connecting "movement of important voices" to the underlying structural progressions of the music. From an even different vantage point, one might look at ideas like the more-recent proposed models of Steve Rings in Tonality and Transformation which focuses on motion of scale degrees in the context of chromatic progressions, sort of like the analysis in the question that tracks the motion of individual voices relative to their scale degree.
I don't know that there's a single umbrella term to incorporate all of these types of theories. A lot of it falls under "transformational theory" as well as "Neo-Riemannian theory." But, again, a lot of theory is concerned with voice-leading, from Cliff Calendar's attempts at measuring general voice-leading perception all the way to systematized Schenkerian analysis.