The William Henry Jenkins I did find in the 1891 and dismissed (see post above) may actually be my man after all. The probate record you provided for his mother Sarah describes his occupation as 'checker'. The 1891 at
Piece: 4485
Folio: 6
Page Number: 14
describes that William Henry's occupation as 'Checker Harbour Trust'. So it is highly probably we are dealing with one and the same despite the two year difference in given age and the oddity over birth place.
This would mean The Butcher's Arms was Jenkin Jenkins' pub until his death when his wife took over as publican (not the son as I had thought). William Henry only became licensee on his mother's death in 1896 - the street directory entry has him as publican in 1900 (compiled before William Henry died that same year). I supposed it had passed from father to son - should have known better as Sarah's younger sister was also a widowed publican.