I often use Share Value ports as a way of maintaining a 'presence' in that location of connections to the input/outputs to a node I'm about to delete. And also as a way of transferring static input values from one node to a new node (although the detachable constant flags feature request would solve that use case).
Also more conventional uses of just wanting to share a value with more than one node, it's a frequently used node. Be handy to just right-click the mouse/stylus on any port and have it inserted on the node or wire that has been selected.
UPDATE: I didn't realise one can already right click on an empty canvas location and create a Shared Value node in that location. So this FR would only improve on that by doing the wiring (and setting the data type where relevant) to the Shared Value output port (and in case of clicking on a wire the input port also) for us.
(sorry for the refusal of this website theme to take a new line, i tried a rule and headers and returns but nothing doin')