A word that covers both chords and dyads?

Hmm. None I'm aware of. There is, of course, the word "interval," which refers to just two notes, but if you start doubling the notes, I don't thing it's still considered just an interval.

There is another situation which is really iffy. In doing four-part harmony exercises in first year theory, in the beginning we were supposed to limit ourselves to using triads. But since there were four parts, at least one of the three notes of the triad had to be doubled. So we were told that it was preferable to double the root. But if that was impossible, the next best choice was to double the 5th. But should one be in a real bind (writing oneself into a musical corner, so to speak) one could triple the root, which meant that you might have three roots and one 3rd. But no 5th. Because the 3rd is a powerful note in the triad (it defines whether the chord is major or minor), we were told to avoid doubling it even if we had to drop the 5th. And it was still regarded as a chord! So, apparently, keeping the 3rd is essential to a pile of notes being regarded as a chord, whereas just the root and 5th is regarded as incomplete.

One some instruments, such as the guitar, multiple 3rds in the chords don't sound all that overwhelming, so they're okay.

The Prof told us at the beginning that for the first year or two we'd be learning the rules, and some of the rules may seem overly strict. But these rules were things that centuries of music had solidly established as "working." The problem, of course, was that if one slavishly followed the rules, one would turn out some pretty pedestrian music. But, he said, don't despair. After we had learned the rules, we would be taught how to break them. But by then, when we broke a rule, we would know why we were breaking it.

Nothing at all wrong with "power chords" or "dyads" if they sound good in what you're doing. That means they're "working," and that's the true test.

Don Firth