Being in or a friend of the Emiya family is the leading cause of becoming a Servant.
Eh, Nasu-multiverse may not be infinite, but it's the next best thing. For any given individual, there's probably a timeline somewhere in the Kaleidoscope where they did something to earn a place on the Throne of Heroes.
It's not that versatile, actually, (Kerry for example can't ever happen except in a history that will never be) but the pseudo-servant thing makes it really easy to servant up pretty much anybody who relates to a HGW in some capacity.