No.
First of all "innate" is completely the wrong term here. Its conditioned, not innate... which is the point.
Second of all I don't feel like writing this dissertation when I should be fixing the AC in the pickup.
Third I generally fucking hated conditioning / reinforcement learning in classes bc it oversimplifies reality.
But here's the brief version.
Pavlovian conditioning is based on the notion of an unrelated thing becoming a trigger for some other action. The. bell before feeding, salivation later when the bell happens. Bell was never intended as the trigger for eating, has nothing to do with eating, its just coincidentally associated.
In this case the God-fosaken Cleanup song was literally made for this very purpose, the signal its time to clean up. So yeah, you hear it and it reminds you its time to clean up. There's nothing coincidental about that.
Now if they gave a cookie every time they finished cleaning, and you smelled those cookies and suddenly hadt he urge to clean, that would be more akin to the idea.
And now that you've ruined this wisecrack I have to admit the Manchurian Candidate is not the same either.
EDIT: plus yes
@Ron is right. Except that sometimes little kids enjoy praise from teacher and that is enough to be the conditioning reward.