I just remembered that in the responsa of Chatam Sofer, Orach Chayim, sec. 83, there is a famous responsum about a mentally impaired child: whether it is permitted to send him to a non-Jewish institution to be treated. He considers whether this is like feeding him directly by hand, and in the end he holds that according to most opinions it is not considered like feeding him directly. Still, as a practical halakhic ruling he recommends not doing so (even though there the issue is curing the child and bringing him to the obligation of commandments).
But in my opinion he is not correct, because in an institution or preschool the staff are the parents’ agents, and giving the food is done as the parents’ agency, and therefore it is considered direct feeding by hand. True, with a non-Jew there is no agency, but when the non-Jew is hired by the Jew it seems that this is still like feeding him directly by hand (in several places we find that an employee is even stronger than an agent, and this applies even to a non-Jew, as with a slave and the like). And in a preschool run by Jews who do not observe the commandments, this is certainly so, since they are subject to agency. His proofs there do not deal with a case where the feeder is hired by the father.

Maybe a preschool is a contractor arrangement rather than employment?

That indeed is something to discuss. But when the contracted job is for a specific action for my child, logic suggests that he is still like my agent. You are not giving him the child so that he can improve an object. And besides, the father bears the obligation of education, and the preschool is doing that for him. Similarly, when I go to a barber and he cuts my hair (performs an action on me), he is like my agent.

A specific action at a specific time.