I’ve read more, from other sources, and I see why the Bishop asked non-Catholics also to sign the document:
He questioned whether someone “can teach what the Catholic Church teaches with zeal and enthusiasm while holding, as they say, ‘in the privacy of their heart’ views that are contrary to Catholic doctrine.
It’s a practical matter, and he does have the right, for a private institution under his leadership, to try to ensure that practically speaking the goals for which he is morally and spiritually responsible are executed.
As a practical matter, it makes no sense for those who oppose (or have reservations about) Catholic doctrine to be trusted to teach enthusiastically and without bias or inerrancy, the Catholic positions – in all matters of doctrine, not just some. Lucky7 mentioned earlier that it would similarly be a practical difficulty for Catholics to teach in a Jewish school, and I think he meant specifically to teach the Jewish religion. Catholics well versed in Judaism (such as I am) could teach a great deal about Judaism until the ministry of Jesus. But what Jews believe about Jesus radically opposes what Catholics believe, obviously. I would not be able to teach “enthusiastically” that the Messiah has not yet come.
For Catholic schools, it goes beyond “teaching religion,” because Catholic teachers & administrators who do not teach religion are asked to role-model, so again, practically speaking, if you are role-modeling something directly opposed to Catholic teaching, you will be (a) unenthusiastic about Catholic positions, and (b) a poor role model, according to the Church, should families discover your oppositional lifestyle. Nowadays people have very little privacy. There is a strong likelihood that private lives would be discovered. Alternatively, allowing duplicitousness in a teacher is something that itself violates truth. (Shielding a controversial lifestyle from knowledge of enrolled families.)
There is a right to privacy in secular employment, such as public schools. Private education is a different matter because of the issue of example, which is built into the very mission of such schools (including non-Catholic Christian schools as well). As a practical matter, those working in private education can expect that lifestyles and beliefs judged oppositional by those institutions, will become a matter of controversy and possibly a matter for firing
as long as the requirements for employment are laid out clearly, which is precisely what this man is trying to do.
I can speak to the other side of the issue, as I no longer offer myself for employment to teach in public schools, given the filth that I am required to endorse and even accompany. I have, i.m.o., wisely chosen to withdraw myself from institutions whose current values I cannot enthusiastically endorse, since that would affect the quality of my teaching and the mission of those institutions to endorse beliefs and lifestyles which are controversial from a practicing Catholic’s point of view.