You're right! If current phones that had a persistent share prominently in the UI came first then this trend might not have happened. But it started with desktop browsers where this concept didn't really exist.
Correct, mainly because "normal" users didn't know how to copy/paste URLs or didn't bother to, supposedly. If desktop browsers had gone in the right direction, they'd have added a "share:" scheme (similar to "mailto:").
In the past, desktop browsers kind of tried to provide such features with the concept of the bookmarklet [1]. The idea was that users would drag&drop a link to their bookmark toolbar to add functionality to their browser. I think it is still used by some Read-It-Later style collection services, but mostly has been replaced with browser extensions.
They managed multiple search engine options just fine and mobile browsers implemented sharing without issues. I don't see a pressing need for user configuration (websites with share buttons don't allow this either), it would have meant potential additional revenue also for Mozilla & Co.
OTOH, they failed to fix "mailto:" for webmail ...
AFAIR toolbars added sharing buttons early on.