I don’t think there’s a way to directly query for observations missing only a date, but I have a workaround that seems to be functional.
Since verifiable observations have a date, a location, media, and aren’t captive, we just have to flip all the switches to get observations missing only a date.
So this asks for observations that are somehow failing the verifiable check, but they’re not failing it due to a lack of photos, being captive, or missing a location – i.e. they must be missing a date.
Note this leaves out sound observations missing a date. Also note observations of Homo sapiens come up in this search, so you might want &without_taxon_id=43584.