Hi Stephen,
Only Allium unifolium is a valid species, one from California that is familiar in the bulb trade.
As you point out, there was a species published in 1923 as Allium uniflorum Larraņaga, but it is considered invalid or illegitimate in every authority that I checked (Tropicos, The Plant List, eMonicot, ITIS, Nomenclator Alliorum). In those days, a bunch of South American plants were published as Allium species, and even some as Brodiaea species, when in fact they are currently ascribed to Nothoscordum, and other genera like Tristagma and Beauverdia, the latter no longer a legitimate genus.
Larraņaga published two other Allium (as did other South American authors), A. pauciflorum and triflorum (not to be confused with valid Allium trifolium), and similarly, these are unresolved names that surely point to other genera like Nothoscordum. There are no true Allium nor Brodiaea species in S. America, only in old taxonomic literature do such names appear.
