I'm a wind player who is interested in constructing musical instruments, and I've been studying acoustic physics and instrument design on my own time. I haven't been able to find the answer, and (knowing that bending changes pitch) I was wondering how the size of the reed might affect the pitch of the produced note. Any explanation would be helpful!
The reed and the air column in the pipe form a single coupled vibrating system, not two separate ones. The ability to change pitch depends on which part is the most stable.
For example most organ reed pipes have metal reeds, and the pitch of the note is controlled almost entirely by the reed. (That is easy to verify, because the pitch of the non-reed pipes changes with the air temperature in the building, but the pitch of the reed pipes does not - though since reed pipes are less numerous and easier to tune by adjusting the reed itself, the reed pipes are tuned to match the non-reeds, not the other way round!). The fundamental pitch of the air column is not necessarily the same as the reed - it may be half the frequency (i.e. the resonator is double length) or have no obvious relationship at all (so-called "free reed" stops like the modern organ Clarinet, or many types of baroque reed stops such as regals, dulzians, sorduns, etc, some with resonators which look very different from a conventional "pipe").
For woodwind instruments, a single reed has to cover a wide frequency range, and the pitch is dominated by the air column not the reed - though of course most woodwind instruments can be overblown to produce several different harmonics of the fundamental pipe frequency. But the reed still has some influence on the frequency of the coupled vibrating system.
The player's embouchure can control the vibration frequency of the reed to some extent to bend the pitch of a note. For "capped reed" instruments (mostly medieval instruments, and non-orchestral instruments like bagpipes which are blown from an air reservoir) where the reed is not inserted directly into the player's mouth, this is not possible, and most such instruments can not be overblown at all, giving a limited range of about one octave.
For brass instruments, the "reed" is the player's lips, which can be adjusted to produce a wide frequency range, and excite many more harmonics of the air column than is possible for most woodwind instruments.