Esrog for children - Obligation?

Igros Moshe (YD II #104)  The common minhag is not to buy an esrog and lulav for a child. 

Igros Moshe (OC III #95)  Question Concerning the obligation of chinuch for the mitzva of lulav and esrog. The gemora (Sukkah 46b) says not to buy 4 species and give them as a present to a child for  the first day of Sukkos  even after he has used it on the first day of Sukkos since that makes it the possession of the child and he is required to be the owner of the lulav and esrog the next day to say the beracha  because of the uncertainty of the status of the day. A child can acquire but not transfer ownership to others. Now, there is a dispute on this issue. Some say  that one can give the lulav and etrog to the child without transferring ownership, and that this suffices to satisfy one’s obligation of chinuch. This is so even though the child performs no mitzvah and actually makes an unnecessary blessing; since for the sake of chinuch it does not matter  whether the mitzvah has actually been fulfilled by the child.  It is sufficient that the child thinks they are performing a mitzvah. Others such as the Magen Avraham says this is not sufficient and he must actually do the correct mitzva. The Mishna Berura says that according to the latter view the Shulchan Aruch says that a parent should hold the four specied together with his son to enable the parent to say a beracha the next day. However in such a cse the child has not done the mitzva since he didn’t own the lulav and esrog and the father did not do the mitzva of chinuch. Nevertheless many other Achronim hold this way. It appears that Shulchan Aruch agrees with this too. Since he says that the father is required to buy an esrog and lulav for his child. The reason that in Europe they were lenient was because of the poverty which prevented the purchase of an esrog anf lulav  for every child. It was not possible even for most adults to have their own esrog and lulav, and instead they took turns using the community’s esrog and lulav and reciting the blessing over it, and were not able themselves to do the wavings of the lulav during hallel either . In fact it is best to be strict in this matter and especially since in America it it possible to acquire them cheaply everyone should buy a set for each child.