I’ve been looking at this, and can’t see what the problem is. Similar searches work OK on other sites I have access to.
Server resources is a possibility, but I can’t see any evidence of that. I tries a general search (using Search menu) for “Common ancestors” which ‘should’ find all those FACT entries PLUS any others where that term just appeared in a note or source or anywhere. It only found 378 individuals, which is not great number. So I suspect it is some other problem completely.
What I would like to try is running the SQL database query that search uses directly on the database itself. Any chance you could organise access to the database for A&D temporarily?