I also have an Arabic book (which was super fun to read on plane post 9/11, oops), that likens it to Scottish's glottal stop, which seems right on in retrospect:
I say letter in quotes because I'm not sure of its status as a letter or not, I think it is a letter but my Arabic game is pretty weak.
Also, what about the word for "thing" شيء it's not placed on the yaa, but written by itself. Or the plural form, things: أشياء has two hamzas in it, only one sits atop an alif.
So that's a thing.