I will strongly say I feel a father doesn't realize just how important his role is in raising his children, especially since society has so often placed it as the mother's job to do such, but I have very much seen how much of an influence a father has in their children both directly and indirectly. Perhaps not so much religiously as just in general.
That parents actions, their words, how they discipline their child, and the morels they teach can strongly effect what sort of person a child grows up to become. That a distant father, or worse, an abusive one, can have a cascading effect.
It's while I hold little sympathy for parents, and yes fathers, who betray their roles and respectability by abusing their children. Robbing them of the trust and innocence the child has.
Being a role model, having a close bond, but yes when it comes to religion things can often be more uncertain. Especially in stricter or more controlling religious households where I have heard from others that such an environment growing up played a notable part in them turning away from religion later in life.
Where a child needs to know they are loved, that they are wanted, and that their parent's love is not something which has to be earned. Moreso, parents need to be willing to sit down and talk to their children, especially when they have questions, and have open discussions. That a child, especially as they become teens, knows they can go to their parents when they needs advice, help, or just someone to listen to them. That they can talk about their fears, doubts, or just stuff that confuse them… and feel safe doing so.