Why couldn't those clever engineers at Canon give me an f22
You have to remember that the aperture number that you see is relative - it's not 'f22' but rather 'f/22', where the 'f' represents the focal length, which you divide by a number to arrive at the aperture diameter. In addition, you need to keep in mind the crop factor - for G16, with its 1/1.7" sensor, it's 4.55. The focal length listed in its specs is 28-140mm equivalent, which means that it gets the same angle of view as a full-frame (35mm film, 36x24mm frame size) camera with a 28-140mm lens, but the actual focal length range on the G16 is 6.1mm to 30.5mm. Finally, you need to take into account diffraction - as the rays of light pass next to the sharp edge of an aperture blade, they get slightly deflected, which results in loss of sharpness - if most of the light goes through open aperture center, this is negligible, but if the center is very very small and a significant proportion of light passing through it gets affected by the edges, you will see significant loss of quality.
Now, for a full-frame camera with a 28mm lens, an f/8 aperture means 3.5mm diameter, which is pretty good, and f/22 means 1.27mm - entering diffraction territory, but still workable. However, with a 6.1mm focal length, f/8 is already a 0.76mm pinhole, and taking it to f/22 would result in just 0.277mm - barely larger than a hair, and quite useless for imaging. On a full-frame camera at 28mm, this aperture would be an f/100 or thereabouts.