To address the original OP, I lean more towards "Families need Fathers" than "Fathers for Justice" when it comes to getting decent rights for Dads to see their children.
When families split, you thankfully have a large number of families where both parents put their own feelings aside and make sure that both sides have access to their children and both provide support.
You then have the two ends of the spectrum - the Dads who don't want to support their children or are refused access because of abuse or one sort or another and also the Mums who don't want to have their ex involved because of their personal feelings and/or have a new relationship.
This is why groups are needed to support the Mums who have to deal with feckless fathers and the Dads who can't get access even when they want it.
What isn't productive is to say "Well my ex was a useless waste of space ergo all men are a useless waste of space and anyone who is trying to get access to their child has probably had it refused as they are an abuser" as it just doesn't work like that and it weakens the argument.
To be honest, the assumption that a man who has had access refused, must have been abusive or feckless is the reason why such groups need to exist in the first place. Some are, some aren't. Women aren't always pure as the driven snow in these cases.