In serve one's/its turn, 'turn' signifies "an individual's time for action, when these go around in succession". But 'turn' must signify something else beneath, as there's no agent.
If 'turn' is synonymizing 'hitch' or 'snag', then how did 'turn' shift to signify this? What semantic notions underlie 'snag' and 'turn'?
Source: Rebecca Gowers. Plain Words (2014 ed). p. 87 Top.
A reader of Milton must be always upon duty; he is surrounded with sense, it arises in every line, every word is to the purpose; there are no lazy intervals, all has been considered, and demands and merits observation. Even in the best writers you sometimes find words and sentences which hang on so loosely you may blow 'em off; Milton's are all substance and weight; fewer would not have serv'd the turn, and more would have been superfluous." — Jonathan Richardson, Explanatory Notes and Remarks on Milton's Paradise lost, 1734