It seems like if I memorized the relative intervals between the notes on each string, it would help a lot with quickly making chords and such. It seems a lot easier than trying to memorize individual notes
Yes, of course! The majority of guitarists have a way to think "directly" in relative terms - often using shapes as mnemonic aid. If you think in terms of numbers of semitones, and have the ability to see shapes on the fretboard in terms of numbers of semitones (and/or intervals like an octave, fifth, etc), then you can easily do all the musical 'operations' you need in your mind without needing to think of note names.
Note that this isn't 'cheating' in any way, or specific to the guitar. It's just as possible on the piano to 'spot' intervals without thinking of the particular note names. It just so happens to be very easy to relate piano keys to note names, due to the octave-repeating pattern of the keys. The guitar doesn't have such a clear pattern (at least, not once you start to move across strings), so it makes less sense to concern yourself with note names when you don't have to.
Of course it is still useful to be able to find named notes on the fretboard, but you don't have to think in terms of them all the time.
if I were to switch to an unfamiliar tuning, would I be completely screwed in that regard?
No more than if you were thinking in terms of note names - those would all move too, if you re-tuned.
One habit to get into is to just work out how a chromatic octave maps to shapes on the fretboard on your particular tuning. Once you've mentally 'mapped' the octave, and once you're thinking in terms of intervals, you should be able to play familiar tunes with only a little extra mental effort compared to standard tuning.