... which is the point. It's all about the spinning top (The Foundation) staying spinning and not falling over...
Premise: You can't expect predictions to stay true over centuries, you need a guiding hand to keep that top spinning and not tumbling over, so you need the Second Foundation to be The Management. Bonus: due to the circumstantial requirements, you get to make them secret and hidden.
Requirement: There has to be tension within a book. There has to be conflict, resolved in a way conducive to literary imperative.
Resolution: Something has to be able to challenge this otherwise-perfectly-set-up system. Something that is not predictable, and which would not be easily adjustable by the Second Foundation; something out of the ordinary. A single man, who really can influence history. The Second Foundation would be prepared for a Napoleon of their time, but they wouldn't expect anyone like the Mule.
Corollary: The Second Foundation is exposed, introducing tension and conflict for the next book in the series. Who wants to be a puppet on a string, after all ?
It could have gone very differently if the Mule hadn't been sterile, or could have taught his skill, but again, literary imperative dictates that "The Good Guys win", so the Second Foundation overcomes the Mule, and the plan is put back on track. Suspiciously well, in fact. The anthropic principle is validated, and a single man did not in fact change history. This time.
"The Good Guys" changes in later books, of course, and the Second Foundation end up losing, but that's quite literally another story :)