I'm fairly confident that they actually do, you just may not realize it. All the first position stuff fits in with the same generic patterns I use to play in all regions of the neck. There are few fixed chords which do not correlate to usable movable shapes, but the root is still easily identifiable even for those—even when the root isn't one of the notes fretted or played.
When I run into unfamiliar chord types, I just derive them by tweaking the dom7 movable shape for the string the target root falls on. This is quite a simple and consistent approach, more consistent than similar derivations from the majors and minors, and certainly faster that trying to work out the names of the notes and how you can fit them together into a usable shape. It's also more reliable than chord dictionaries, particularly the online ones. It's another powerful argument for learning generic patterns rather than picking everything up piecemeal as special cases.
In my estimation, the little bit you gain in tone by sticking to the lower neck is dwarfed by what you lose in musicality. I'm not saying you have to use the whole fretboard, but it's a bad idea to restrict yourself to first position.
I find "Hotel California" to be a great example in support of learning the fifths pattern. Fourth progressions are just fifth progressions in reverse, at least in terms of the root movements. In addition to the fourth progressions you identified, the chorus is entirely composed of fourths and fifths in relation to the two main relative tonics, A [for minor] and C [for major]. The first modulation is to a key a fourth up from the main tonic, A; go another fourth to get the root of the first chord in that subsequence, then backtrack a fourth. Then drop those two chords a whole step, which lands you on the tonic of the relative major key—note that tonic position for when you get to the chorus. Using only the fourths/fifths fretboard pattern and some root movements by whole steps, I was able to play this song in several keys, without even having to think about the fixed chord names. There's your "Circle" Power in action.
The moral is that there are often more fourth or fifth relations lurking in unusual progressions than may at first be apparent, and if you recognize them, the 4/5 patterns can simplify both remembering the progressions and playing them.
Furthermore, all the diatonic roots in a key, be it major or minor, are chained by fifths (or fourths, depending on whether you think of movement down or up) in this order: 7 3 6 2 5 1 4 (b7 b3 b6, for minor); this is also the typical order followed in fifths progressions. 7 is only a half step below I, 3 is only a half-step below 4, b6 is only a half step above 5, so there are several quick ways to follow the fifths pattern to get any needed degree, approaching it from different points. There are other ways to do the same thing, but they may not follow as compact an arrangement as the 4/5 pattern can describe, and may involve more conscious mental work. I now mainly use a set of five "root cluster" patterns instead—a bit more random-access—but I derived them all from the 4/5 pattern initially, and still find that pattern faster in many cases.
On a circle diagram, you'll see the flats sequence BEAD-GCF repeated around the circle counter-clockwise (the direction fifth progressions follow), first in sharps, then naturals, then flats. So there in a nutshell you have the entire sequence of fifths in a form you can quickly memorize—and extending a little farther than the circle does, with no naming confusion where the circle eats its tail (those enharmonic equivalents). When you see subpatterns of the BEAD sequence, like Am Dm7 G7 or Cm7 Fdim Bbm, you know you're dealing with fifths progressions. Fourths or "back-cycling" progressions go the opposite way.
If you know the diatonic chord types expected on the different scale degrees, the easier it is to spot when things conform to expectations and when they do something different: chord "tensions" (coloring), secondary dominants, modulations, borrowed chords, tritone substitutions... This helps you understand the harmonic structure and also provides more hooks for your memory to grab onto.