1. Question: If the Be’ur Halacha (in 657) says according to the Shulchan Aruch you must buy a child a kosher set of lulav and esrog because proper chinuch is to teach him how to do the mitzvah properly, but a child can’t do it properly anyways since in siman 60 we say that mitzvos must be done with kavana, proper intent and a child can’t have halachic kavana, so if he can’t perform the mitzva anyways then why are we required to buy him a kosher set?
A. There is a famous chakira about what it means that mitzvos need proper intent: a. Does the intent make the action, meaning that it’s only considered an action of a mitzva with the intent but without the proper intent it is just a regular action not counting as an action of a mitzva.
b. The action by itself is considered an action of a mitzva even without proper intent, but the reason why we need the intent is for the fulfillment of the mitzvah, for there are two parts to the fulfillment of a mitzva, the action of the mitzvah and the proper intent for the mitzvah.
Answer: If you say like side b then it makes sense because using a kosher set of lulav and esrog allows him to have a proper action of a mitzva, and his mind is saying he would like to perform that mitzvah, it’s just that it’s a mind of a child which can’t have proper intent but the parent is still doing all that he can to show the child how to perform the mitzva properly.