Statepower is no more to be used so as to handicap religions than it is to favor them.
This reflects the unanimous decision in Cantwell v. Connecticut, which was decided seven years prior to this case and likely set a precedent. The Fourteenth Amendment, specifically the Equal Protections Clause, prohibits discrimination by local, state, and federal governments (solified by the Cantwell decision) on the basis of religion. With transportation being a public service and religious schools being private institutions, it can be expected that transportation to a private school would be less profit-oriented just as transportation to a public school would be.