Seeing that the search has been bricked for a while, the idea of a link to another window or tab page for a Google search in Stearnvault might present as a (temporary) workaround. Modifying the current search boxes is sure to be a thing, Sod’s Law has it that as soon they are operational, the native site search will return.
We may need to sign up for Analytics and/or Search Console. I think the latter is what we would want for that but Google has a ton of tools. There is also a WordPress plugin that helps to integrate with them if we go that route.