As far as some states never mentioning or listing the father's name,
why would the priests try so hard to protect his identity?  Double
standards, perhaps, or is there a Catholic or cultural thing I'm not
understanding?

Let's see if I can get this right:  The priests would not necessarily be protecting the identity of the father, more like the priest would have not sanctioned such an off-spring, and so the woman and child forever was "labeled" because of their parent's indiscretions.  However, on the other hand, where the Catholic Church and government often acted in unison - it meant that the illigitimate child and the mother never would have an opportunity to claim any inheritance from the father's family.