To me, I think it's often better to think of these things in the time domain. "the spectrum is composed of peaks at multiples of a fundamental frequency of 1/T" is equivalent to saying "the waveform (in the time domain) is periodic with period T". Anything that manages to produce a periodically repeating waveform will be perceived as having a pitch. Savart wheels (https://en.wikipedia.org/wiki/Savart_wheel) don't have anything that is vibrating à la strings, air columns or reeds, yet still sound pitched. So as long as the reed manages to make a periodic disturbance in the air, the sound will have harmonic overtones.
Even for plucked strings, it's just as good to think of what is going on is something like this: the string is plucked at a point, two disturbances, one in each direction along the string, propagate away from it, they bounce off the nut/bridge and so on. The amount of time it takes for the wave pulse to make a complete round-trip is the fundamental period. It's that we get this kind of periodic behavior that gives the sound its pitch (of course you won't have the pulses run back and forth along the string without change unless the stationary modes have frequencies that are integer multiples of the fundamental...) Anyway, I find it useful to think this way in order to not get too hung up on the fact that the vibrational modes of reeds don't have harmonic overtones, and, as far as I know, the higher order vibrational modes aren't relevant for the sound production of the reeds.
I wonder if anyone has tried exciting a free reed without using airflow. One could take a steel reed, stimulate it with a magnetic field at the correct frequency and see what a reed sounds like without airflow through the reed frame. I'm still unclear whether we "hear the reed vibrating" or "hear the reed chopping the airflow through the frame".