Does donating to an orphanage in its entirety, with the intention of sponsoring all the orphans within it—for the purpose of ensuring equality among them and avoiding jealousy from unsubsidized children—count as the reward for sponsoring an orphan?

Paying money to an orphanage fulfills the sponsorship of orphans and earns the companionship of the Prophet in Paradise, if the money is sufficient to meet their needs. If it is not sufficient, it is considered a participation in sponsorship, and one is rewarded for it accordingly. It might even carry a greater reward than sponsoring a single orphan, especially if the money is paid to avert a harm.