The alteration of the three and two letter ICAO designator completes the process of the name change from Jersey to British. So as far as the public is conserned the Jersey name no longer exists.
It was obviously confusing for them checking in to a British European flight to see the flight number listed as JY***. So as a marketing and
PR perspective it is a great help.
Much work was put into a new callsign to avoid this RT confusion however the powers that be decided to keep the old one. "Beeline" (which I don't think was ever used by us not even for half a day)was the odds on fav however as you point out would probably have upset the older BEA / BOAC / BA chums.
As for the confusion well BA use speedbird as many other airlines use callsigns that do not match up with their company name.