(Exhibit "A") http://youtu.be/JgFt6QFzfEM?t=1m7s Hear how the fundamentals & overtones align progressively as David involves more drums, & even eventually the fundamental of the bass drum (use good headphones or similar). Given enough drums, something that most would consider a "chord" is entirely possible.
My take from a drum designer's perspective, it's all in the management of overtones, & yes, they can be
managed within the instrument design. Broadly, you have two groups of overtones - lower overtones (but obviously, still above the fundamental) that are typically static features, & errant overtones (usually high, & irregular in their manifestation). A high quality instrument is designed such that the errant high overtones are largely engineered out. That leaves a judgement call on the balance between the remaining overtones & the fundamental. Drum designs that reduce overtones to near audible zero are entirely possible, but regarded as not very satisfying to most players. Too thin, too one dimensional, so a mix is used to "fatten" out the tone.
Most drum sets have a fairly diminished fundamental. The example in the first video is more balanced by comparison. In this example however, errant high overtones are completely eradicated, but "fixed" overtones are encouraged even more than the first example. This changes the bias more towards a "tuned" instrument
(exhibit "B") http://youtu.be/XojpGAmH4NI?t=44s
BTW, piano notes are similarly composed of fundamental & overtones, but as there's fewer variables, the overtones are typically closer to (but not necessarily the same as) the fundamental harmonics. Once sympathetic resonance of the other strings, frame, & body of the piano is involved, a larger & often more "eclectic" range of overtones are at play. The key difference between piano & drums in this regard is repeatability / consistency, & the bias between fundamental & overtones. As I mentioned earlier, it's entirely possible to bias drum design heavily towards a super dominant fundamental, & in such an example, the difference between piano & drums are fairly small. It's just a question of number of notes & practicality.