I managed to speak to the Google Adsense for Search team about using my Google Adsene Account in the SiteSearch in IE7, suffice to say I was correct in my first guess: You are not allowed to use anything but their search boxes. I think this is a bit limiting, but the person I spoke to said they would pass it to their product team for consideration.So in summary, using IE 7's open search provider with Adsense is a no no.
I lead the Chrome Developer Relations team at Google.
We want people to have the best experience possible on the web without having to install a native app or produce content in a walled garden.
Our team tries to make it easier for developers to build on the web by supporting every Chrome release, creating great content to support developers on web.dev, contributing to MDN, helping to improve browser compatibility, and some of the best developer tools like Lighthouse, Workbox, Squoosh to name just a few.