Sadly no words in the article about the physics of sympathetic resonance, which is a great part of the rich sound of acoustic instruments. The 'secret' of the missing fundamentals is just there.
Gently depress and hold the keys C, E, G, C' in any octave near the middle so they don't sound. That takes the felt off the strings.
Now with the strongest finger on your other hand, wallop the D next to the initial C as hard as you can while still holding the C major chord keys. Release the D key and listen...
The harmonics damp out at different times and leave certain overtones ringing. A high pure tone is last on a well tuned piano. My joanna gives an interestingly rich range of organ-like tones first though.
Now imagine synthesising all of that dynamically for arbitrary chords, note length, pedalling &c and you have your problem space!!
Domain relevant accoustic experience: track down a recording of Charlemagne Palestine's Strumming Music and listen to it on good quality headphones. His original recording (the one with the subway train going under the building as the last notes ring) was performed on his preferred Bosendorfer Imperial piano which has an extra half octave in the Bass end of the keyboard. No music written for those notes, but they don't half change the sound.
The problem is not that the piano has too many strings, it's that the notes are fixed. The same is true for any string instrument with frets, like a guitar, or wind instruments with fixed holes, like a flute.
The video also refers to the fact that equal temperament allows playing in any key, but doesn't really explain why that's important or what the tuning has to do with it. (Short answer, before equal temperament, different compromises were made in which notes were out of tune with respect to which others, and the schemes commonly chosen only allowed some of the keys (as in "c major" not the physical key) to be sufficiently in tune to be usable.)
As I understand it, people were aware of the problems of the temperaments, but a temperament is a technology. Each of the old temperaments came with an algorithm, that your regular Joe musician could use to keep their own instrument in tune. This in turn was needed because instruments didn't stay in tune for very long.
Equal temperament requires an expert, which in turn requires an instrument with stable tuning -- the modern piano.
Wind instruments actually have no straightforward temperament, but are just as close as possible, and the musician is expected to bend notes as needed. For all practical purposes, an orchestra is an un-tempered instrument.
