Appliances I have seen or repaired that do not have a primary fuse - only light fixtures and 1960's car battery chargers, vintage tube radios.
Possibly a few old wall warts, although I'm not sure as I didn't unwind the transformer looking for a thermal fuse. Even in the 1980's there were thermal fuses in consumer electronics (small tranformers). I used to repair clock radios, ghetto blasters, stereos, TV's etc. in a shop, not casual numbers.
In low <~20VA transformers, fine primary copper wire does end up being a fuse at the bobbin lead-in. I have seen two melt there, say smaller than
#28 AWG. Inductance leading to an arc in the bobbin, can't be a good idea with this method. It's odd the UL standard does not allow it but the IEC standard seems to:
IEC 61558-1; 15.3.5 intentional weak part. "may be an... inaccessible weak point in a winding."
UL 1585; 18.2 "Crossed or reduced cross-section conductors shall not be used as a protective device. A nicked conductor is a form of reduced cross-section conductor."