Is the CAGED system the main way for learning the guitar's fretboard?
I suppose you could answer "yes" because it seems to be popular. In my lessons I certainly learned
C A G E D chords first. But I hasten to say I wasn't told this was "CAGED" and I didn't learn the chords in that sequence - it doesn't make much sense - I probably learned them
E A D G C in descending fifths.
I dislike mnemonics like "CAGED" and simple systems to claim to be the key to unlock everything. The first time I heard "CAGED" I thought it was silly. I mean really, if you keep to those common chord, how many songs can you get through without
F? Of course
F can be played as a CAGED form higher up the neck out of open position, a barre form will be typical. You can look at this as an example of how everything relates back to CAGED... or you can see is a the being of propping up the CAGED system with lots of modifications to make it practical.
Whether it is the "main" way someone first learn the fretboard it certainly isn't the only way, and that is the important thing.
Personally I learned to identify octaves, fifths, fourths, and third, sixths, and sevenths on the fretboard as a basic way to find my way around. Fifths and fourths will give you a grounding in your primary tonal degree locations and roots for progression by descending fifths which is an important harmony fundamental. Thirds, sixths, and sevenths will help you fill in the modal qualities to build and modify chords. Basically a combination of interval identification, diatonic scales, and tonal harmony patterns.
do you do a combination of them [CAGED boxes]?
For the system to be practical "yes."
But you should watch out for when CAGED goes from being an aid to an obstacle.
To learn tonal harmony (diatonic plus standard chromatic harmony like secondary dominants, borrowed chords, etc.) you will have to superimpose those pentatonic boxes in various ways to produce a diatonic set.
Is there a benefit to learning tonal harmony as a bunch of superimposed pentatonic scales? How is it better than just learning the diatonic patterns that underlie tonal harmony?
If learning tonal harmony isn't a goal, CAGED may be sufficient. If you want to go beyond a pentatonic focus, don't get CAGED in. :-)