It seems that 1/2.3 sensors would produced fields-of-view that are tiny, even at the shortest focal lengths.

You're not completely wrong — you're just underestimating the limit.

The relationship between sensor size, focal length, and field of view is simple — see What is "angle of view" in photography? for details, or simply What is crop factor and how does it relate to focal length?.

A 1/2.3" sensor has a crop factor of around 5.6. That is, to have the same focal length as a full-frame camera, the lens needs to be 5.6× shorter. That means to give the field of view equivalent to a 24mm wide-angle lens on a full frame camera, you need a lens of about 4.3mm. The widest-angle consumer camera with a 1/2.3" sensor I'm aware of is the Canon SX60, with a superzoom lens which goes from 3.8 to 247mm. At the short/wide end, that's equivalent to the field of view of a 21mm lens on full-frame.

So, as you note:

And yet cameras with 1/2.3 sensors seem to be able to produce photographs at wide angle setting that have resonably wide FOV.

Yes, "reasonably wide" is possible. But wider than that seems to be an engineering challenge not commonly met.