The Mishna Shabbos 121a states that if ones house is burning down on Shabbos and a Goy wishes to extinguish it for you, you need not stop him. However, a Jewish child must be stopped. The Gemara explains that the Goy does it for his own interest, but the child does it for his father.
From the context, it is obvious that there is no Heter if the child will benefit as well. The source in the Mishna and Shulchan Aruch is his house burning down too, and nevertheless it is forbidden to let him extinguish the fire despite the extreme loss of money and discomfort.
In conclusion, I cannot take responsibility for the words of Rabbi Cohen and Rabbi Leff, but the Gemara, Shulchan Aruch and basic Poskim all rule that any child old enough to understand when his parents are happy with his actions or upset, is forbidden Min HaTorah to do Melacha for the parent even without an explicit instruction or thanks. I think that any parent will agree this occurs well before the second birthday.