Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
We ask Allah to guide all those people who are sponsoring these programs. May Allah grant them success and bless them. Teaching Muslim children is an utmost important issue and we are asked to do take care of it, especially the Muslim children living in western countries. No doubt it is a part of the Mosque's mission and of its role in Islam. As long as the Mosque can carry out the sponsorship of the school financially, then it should undertake it to receive a maximum number of Muslims' children, and this work should be the most important part of its activities.
We also approve charging a nominal fee to invest in this field, on those people who can pay it and exempting those who cannot pay. Or, you may make it optional: those who want to pay can do so, and those who do not are free to do so. At the same time, we believe that it is a duty to call the Muslim community for donations, alms, for this cause or the like.
On the other hand know that giving donations of money and alms (if not from Zakah as you mentioned in your question) to the people who are not needy or poor is permissible.
Allah knows best.
*Helped by fatwas of islamweb.net