It is mildly interesting that the ordering of the chords, when I start with the F maj, is EDCAG, so it spells a word by circular-shifting the letters 2 places left.
You are correct, that is only mildly interesting. It should be obvious to most that everything on the guitar repeats itself and is ordered.. the 12th fret on each string basically starts over again from square one. Same for each string.
So we know the guitar is fix and repeats and that is mildly interesting and pretty easy to understand.
Now... dig a little deeper. Each SHAPE also has an order. A musical order. For instance the E SHAPE from low to high spells 1 5 1 3 5 1 on the strings which is a major chord.
so you have 5 SHAPES and 5 ORDERES. Each order is different. For instance the A SHAPE is ordered low to high.... 5 1 5 1 3 5 ... still a major chord but the order of notes is different.
A Major Triad chord is spelled 1 3 5. The first inversion is spelled 5 1 3 hmmmmm that's the same as the A SHAPE.. Now look at the C SHAPE... it's order is 3 1 3 5 1 3 ... the second inversion of a Triad is 3 5 1... notice the 3 5 1 sits in that C SHAPE
So0 the SHAPES help you see inversions rather quickly. Now what if you combine a couple of shapes? What if you need a chord other than a triad. Like something with a b9 or a #5... can you find it at any given point on the neck?
Now CAGED becomes a bit more than simply mildly interesting. It's the road marker system. It's like waypoints on a gps.
Practical example.
Chord Prog. Am G F E7
If you play root notes on the b string descending you will notice that the F note falls into the A SHAPE for a Dm chord. So over that chord you could play your old familiar D minor pent at 10th fret ( which also happens to be F maj pent G SHAPE )
It basically becomes an endless web of potential. For instance rather than use it as "go to" device, you can find yourself playing certain melody notes and simply say... ok, what do those notes form in reverse as a SHAPE or partial SHAPE and thus what chord might that be? You will find it is not always the chord that you are playing over. As in example above the chord playing was F Major but the melody could be visualized or conceptualized out of Dm.... or maybe even a Eb9 because that F gets some resolution from the E chord and it then resolves to the A.
So if you got to thinking about all that stuff you might want to know how to play a certain E Scale at that certain location on the neck near the 4th fret.
IOW, once you dig a little deeper it's pretty much only limited by your imagination.