I have been playing around with the search in an attempt to improve it a bit.
Turns out the search within a topic seems to be borked. Some obscure stuff in the back end coding that I don't have time to track down at the moment (it's a jungle in there). So, that option is gone.
Search within board seems to be ok, so that stays.
What I've also done is to enable Google search of the whole domain. This will only pick up public content that is visible to guests/lurkers, since that is all that spiders can access and index. It's fast though, and may be handy for a lot of things.
It opens the standard Google search results page, which is fine IMO (I'm not going to bother to try and import the content from that into the forum templates).
Lurkers/guests get the Google search by default, and don't have any other options.
ETA: Oh and if you're logged in on mobile you get what's in the pic below. Figured you might as well have the same options as desktop, but needed to save space.