Your question is assuming they're not already locking down as many of the 50 other ways as possible. Yes there are other bad things but that doesn't mean we should add #51 to that list.
I suppose it tells them you searched with the address/search bar instead of navigating to the search website. But I'm not sure why you would care about that.
Can you describe how? What information is that query parameter providing to an attacker that is not being provided in 50 other ways?