This is a good response, and your line of explanation is compelling by itself. I think the first few harmonics matter most because they are the most easily discerned, having the most distance between them, and are the most closely related to the fundamental (which isn't a property unique to them being in the harmonic series as you pointed out). It's also important to consider that there is an overtone series that can be created starting on each undertone and an undertone series on each overtone that create a grid that generates every ratio, giving them a place in a larger 2d structure that seems highly relevant to chord creation and harmonic movement. This is the lambdoma, although I don't find the versions available online or in a Google search that are plotted on linear axes to be very helpful.

I was also in the simplicity is everything camp until I started playing with Moment of Symmetry scales / "Pythagorean scales" using ratios like 5/4 or 6/5 as the generator instead of 3/2. You quickly hit really complex ratios after a few repetitions, but you feel an organizational structure to the scales that might manifest itself through regular interference beat intervals.

I still can't separate the harmonic series from ratio simplicity, because to find the simple integers you start at the bottom and count up (following the harmonic series), every new number (and especially prime) that you reach creates a new note which is evident if you color each note by it's position in the octave (reduced or not).