The circumstances of the death, if correct, would have led to an inquest. For inquest deaths, the coroner is always the informant who instructs the registrar to register the death and that can't happen until after the inquest is concluded - historically that wouldn't usually mean much of a delay but now can be months (or occasionally years) later.
The registration will appear in the usual death register though - so perhaps the family story has got confused over the years ( or was invented in the first place).
I am inclined to believe the folklore, which came from my cousin to whom Charles was also a great-uncle, her mother having also grown up in Liverpool. Charles appears in the 1901 census, unmarried, still living at home; also in the annual Liverpool trade directory until 1903 with the Lancs & Yorks Assurance, where he was a middle manager, as confirmed by the archivist at Aviva, the ultimate inheritors of the L&Y.
Of course there are all kinds of possible explanations, such as emigration, change of name, inaccurate transcription, etc., but I would have thought some kind of death record might be found, in later years if not then ?