So that’s my modest proposal. Extend the list of possible values for the type attribute on the button element to include “share” (or something like that). In supporting browsers, it triggers a very bare-bones handover to the OS (the current URL and the current page title). In non-supporting browsers, it behaves like a button currently behaves.
I’ve contended with the Web Share API for Unmark‘s PWA enough to know that this would be excellent to see happen.