Bill,

I believe you’ve come up with an original solution to an old problem! I’ve been
reading OLDTOOLS for about two decades; the only similar repair I can recall,
and this is VAGUE, drilled the main body of the handle and attached a new piece
of wood which was then turned to a taper.

Your solution seems better. Your spigot goes to the bottom of the socket. The
stress of a hammer blow does not go through the glue joint. (At least, not if my
understanding that the spigot includes the tip of the original taper which still
rests on the bottom of the socket.

Finding mushroomed socket chisels is extremely common IMHE.  Some were sold
without handles; certain tradesmen beat on the sockets from day one.

John Ruth
In windy Metuchen, NJ