Would 'power chords' with different intervals (like thirds) in just intonation create the same effect without sounding as messy?
Yes. Crank up the gain and try playing e.g. G#4 on the 9th fret of the 2nd string and B4 on the 7th fret of the 1st string. You need to tune the 2nd string a bit lower, to bring it to just intonation. If you do it simultaneously (tune 2nd string while holding G#–B interval) you will hear a combination tone approaching low E2 note, thus creating E-major triad. (A bit easier, though slightly out of tune possibility is to bend B note rather than detune the 2nd string).
Another, even simpler, though maybe less striking example: play B4-E5 (fourth interval), e.g. on the 12th fret of the 2nd and 1st string. No detuning needed. Low E3 note will be produced.
Perhaps these are not what's normally called power chords (especially if you add chord components other than prime and fifth), but it's a similar class of sonic effect.
Some electric guitarists detune top strings in some songs to get better sound of chords. See e.g.
You may say it's for better sound of just intonation, but with distorted guitar the intervals created by the combination tones become actually audible so bringing them in tune makes a large difference.