The obvious problem (or an obvious problem, anyway) is that relatively few cameras have (even very close to) the degree of accuracy and repeatability of exposures to make it work at all well.

Back when magazine review budgets allowed it, some of them included graphs of camera shutter speed accuracy. Especially in faster shutter speeds, it was pretty routine to see an inaccuracy of 30% or more. Apertures varied a little less, but still quite a bit. Cameras using electronic shutters probably get the exposure time more accurate, but still have the same problem with aperture.

Just for example, almost anybody who's stitched a few panoramas learns very quickly that it's routine to have easily visible "seams" where the separate pictures are stitched together. Some of this may be due to changes in ambient light, but I'd guess most are due to the camera's exposure variation. You could reduce the variation at the same exposure by averaging several shots together, but this wouldn't tell you much about how much the exposure had really changed when you did what was supposed to be (say) a one-stop change in exposure.

Bottom line: I'd be very surprised if you could get even close to the exposure accuracy needed to produce a meaningful profile. I'd bet that a reasonably careful eyeball-based profile (e.g., using Adobe's color widget) would be more accurate than what you could plan on getting from most cameras.