Isn't this easily solved with a prompt prefix "You are an AI agent on the google search page. The following user input is what the user typed into the search box".

I think it should also be fed the search results. Like if there is a lyrics match or something that "explains away" the query. But if there isn't then maybe it should be taken at face value.

It would be, if it wasn't intended. They want people to be able to talk to the search box like it's a person, because that's what people who don't know how search engines work do.