The father, as ruler of his family, has responsibility to train up his children and teach them Yahweh's love and purpose.

Proverbs 22:6
Train up a child in the way he should go: and when he is old, he will not depart from it.

Ephesians 6:4
And, ye fathers, provoke not your children to wrath: but bring them up in the nurture and admonition of Yahweh.

However, that becomes difficult or impossible if authority is stolen from him. There are many ways his authority can be usurped or stolen. The most common is through court orders in divorce proceedings. A family lacking the authority of its leader will lack power and effectiveness. That doesn't imply that a motivated mother can't do the job. But is will be more difficult and the family unit will suffer and this is exactly what Satan and his proxy minions want.

Another common cause of decreased paternal authority is through contention. Children easily know if/when there is a lack of unity between their parents. Fighting between a husband and his wife is a "kingdom divided" against itself.

Luke 11:17
But he, knowing their thoughts, said unto them, Every kingdom divided against itself is brought to desolation; and a house divided against a house falleth.

For a family to succeed, it must be united, "one flesh".

Genesis 2:24
Therefore shall a man leave his father and his mother, and shall cleave unto his wife: and they shall be one flesh.

Frequently (all too frequently) father's are removed from the family because they are incarcerated. The United States has one of the highest incarceration rates in the world. Many (especially fathers) are in jail for victimless (non-existent according to Yahweh's law) crimes. This is another of Satan's plans to decrease the power and influence of the family.

A family united in Yahweh can maintain its power and authority.