So does this mean they were okay to force all staff and students who didn’t have a specific religious reasons?
Universities have always had the authority to mandate certain vaccinations. Their COVID policy was unconstitutional because they did not allow for an individual’s religious objection to the vaccine to be used as a means of obtaining an exemption. The 1st amendment does not base a person’s freedom of religion on whether they are officially members of a sect. Therefore you are not required to base a claim that your religious freedom is being violated on what your church believes but on what you believe.
For example, a nurse refuses to participate in transgender surgery based on her belief in the Bible teaching that God made two sexes. However her administration finds out that years ago she was baptized in a faith that now supports transgender “rights” including access to surgeries. Her request for an exemption is refused because (although she left it) her official religious affiliation does not object. That is the type of situation the university policy leads to.