web: fix event propogation in search-select wrappers
Two different patches, an older one that extracted long search
blocks that were cut-and-pasted into a standalone component, and a
newer one that fixed displaying placeholder values properly,
conflicted and broke a relationship that allowed for the values to
be propagated through those standalone components correctly.
This restores the event handling and updates the listener set-ups
with more idiomatic hooks into Lit's event system.