I too wouldn't consider a teacher to be a holder of public office; however, in reality, how many schools do you know of that put the particular ("an atheist for a main scale maths teacher role, or even to be a lunchtime assistant.'), ideal into practise - I know of none. The important thing would be how they do their job and any religio/non-rel would be secondary as long as they keep it private.
Not true - many adverts for vacancies in RCC schools indicate specifically that the candidate should be a practising catholic, and while in lower level positions there may be some flexibility it is clear that a candidate who is a practising catholic will be preferred.
For lower level positions there is usually a statement about being a practising catholic or upholding the ethos of the school, which all seems fine and innocuous (because of the 'or'), except this is considered to apply not just during the performance of their duties in the school, but also in their private lives.
So from the Catholic Education Service's own official documentation:
Employment of teachers in Catholic schools
“There are also substantive life choices which are incompatible with the teaching of the Catholic Church and which may be detrimental or prejudicial to the religious ethos and character of a Catholic school. Some examples of these would be:
• formal apostasy from the Catholic Church ...
• a Catholic contracting a marriage in a non-Catholic church, registry office or any other place without dispensation from canonical form;110 or contracting a marriage where one or both of the parties have been previously married (and whose former spouse
is[are] living) without the former marriage(s) being annulled or declared invalid by the Church;
• maintaining a partnership of intimacy with another person, outside a form of marriage approved by the Church and which would, at least in the public forum, carry the presumption from their public behaviour of this being a non-chaste relationship; and, where such a presumption in the public forum is not repudiated by the parties within the relationship.”
So in effect even if someone is not a practising catholic, thinks they are OK under 'upholding the ethos of the school' the way that ethos is defined extends to their private life and specifically excludes atheists.
So you can have your contract revoked if you get divorced for example. Or (a real case I know about) have a job offer withdrawn because the applicant was separated from his wife.