Your question is beautiful and I think there will be more science and research behind it thanks to the advent of the Spectrogram (where you can visualize all the frequencies engaged for a particular sound).
Timbre and harmonic invokation are synonymous to me because when you trigger a string on a guitar or piano, or engage a flute's resonant chamber, you are engaging a whole system which somehow (miraculously) produces sound.
Can we consider timbre to be a specific type of chord?
Certainly. Look at a spectrogram when you play any note and you'll see that many frequencies are engaged.
If i use an instrument with a "rich" timbre, with lots of overtones, then isn't the sound more prone to "cluster" if i play a set of notes(worst if near each other)?`
Well effectively, it's hard to say, some get more pronounced, some frequencies get "buried" in the background. In general, the harmonic engagement of a sound is spaced such that there are many higher (and some lower) octaves engaged. When you start making music that is more than an octave or two apart, the lack of harmony or dissonance that is readily apparent at adjacent notes becomes pretty unnoticeable. Consider a major 7th, it's not even a whole octave away but still sounds pleasing, but if you played a B next to a C, you would hear dissonance. Almost an octave away and it becomes pleasant and beautiful. So the harmonic engagement / distance that they exhibit is a big deal, and that lack-of-same-octave-overlap makes it so that the sounds are just fine.
Could there be lots of clustering? Well yes, consider a piano with the sustain pedal engaged. You start rumbling on a low note and go a few octaves higher.. Eventually the whole resonant body will go wild and you'll get dissonance, but that can also be a desired effect.
So your question,
harmonic implications of an instrument's timbre
Is best examined through a visual representation, like that of a spectrogram. Soon there will be tools to analyze intervals clearly. As far as I know there is no in-depth study of these things, but in the end it comes down to listening.
Just wanted to chime in with my thoughts so far, as it's been a topic of interest for me lately as well.