If you play any piano, or even if you don't, I'd highly recommend learning a little about jazz piano voicings. There are lots of good books on the subject, or you can just do a web search for the term.
You'll find that even on piano, where all sorts of big two-handed voicings are possible, it's still quite common to play very spare 2- or 3- note voicings. These voicings can indeed be ambiguous--e.g. is (E3, A3, C4) a C6, an Am, or an Fmaj7?
But they still sound good, because context (the melody, the bass line, the chords that come before or after) allow the listener to hear the harmonic function. In fact voicings which include every note of a complicated chord can often sound too muddy and confusing, especially if they're played in too low a register.
By the way: "Let's take a 13 chord, like A13. Its semitone classes are 0, 4, 7, 10, 14, 17, 21." That's not quite correct. You probably heard that dominant 13th chord is defined by stacking thirds from the mixolydian mode. But in fact you'll almost never hear both the major third and the perfect fourth in the same chord. It would be more common to raise the fourth (replace that 17 by an 18) or leave it out entirely.
Finally, I don't recommend talking about "semitone classes". I know it seems like the simpler and more logical system, but the fact is that any educated musician will know immediately what you mean when you talk about a "thirteenth", but most will need to stop and do a quick mental calculation to figure out which interval has 21 semitones.