Well that's not Catholic.

The father is the patriarch of the family, and has authority over everyone else in it. Then the mother over the children.

The mother becomes holy by practicing the virtue of obedience to the father. The children become holy by practicing the virtue of obedience towards the father and the mother. The father becomes holy by properly exercising his authority over wife and children, and the mother over her children.

This sort of thing works well for minor decisions, such as, whether the family is going to eat dinner then pray the rosary or if the family is going to pray the rosary then eat dinner.  But it is absolutely insane to suggest the children have a voice in major decisions.  There is a reason children are in the care of their parents:  They haven't learned all they need to know to make such decisions.  

I'm not saying that there is anything wrong with asking for the opinions of the children in many matters but I can not imagine (being in the midst of raising five children) giving them any real authority in those family matters.  One or two kids have, on occasion, actually contributed to the decision making and have been able to articulate valid issues to consider, but this is not the rule.

It would seem to me that only parents (especially the father) of a family who just don't want to take any responsibility for any failure would even suggest such an arrangement.

The OP sounds like she is talking about someone else's family.

In such a case, I would not call an outsider a prudent judge who can dispassionately observe a family. Families get a lot of flack from outsiders based on very little. Maybe even a member of the family thinks their family is run that way or they present themselves as such, knowingly or unknowingly. Can an outsider truly know and understand the family well enough to make a statement.

Does the family truly put into practice liberal ideas that directlly rebel against the teachings of the Church regarding the family? I think that would be pretty hard to prove.

I find it hard to believe any family runs like a complete democracy.