How do men remain leaders in a world where women became strong?
By adjusting their concept of what it means to be a strong man.
A leader can protect anyone under their wing, while bringing out the best in them. A leader cannot be a leader by putting down, downplaying and oppressing the skills, natural essence, potential, and talents of others. If they have to do that in order to maintain leadership, they are weakening the group as a whole, and weakening some individuals in that group.
That means that if a woman is good at nurturing children and bringing people together through empathy and building trust and cooperation, you don't put her down for being emotionally attuned, in order to elevate more masculine qualities like overcoming pain.
If a woman is good at problem solving and engineering, you encourage her, rather than encourage her to abandon that interest in favor of a more "feminine" role.
It means not tying your ego in being the best in everything, but instead at being the best at creating a strong team, and at using your aggression to protect its individuals when necessary.
This ideological change is not easy. I believe men and women have different biological inclinations and disposition toward a certain temperament and sets of skills. Men were born to fight people and obstacles, and they need to learn to overcome adversity in order to do that well. Unfortunately, they learned to differentiate and define themselves in contrast to feminine traits, which means that in order to elevate masculinity, femininity has to be put down. It has to be put down, even though the feminine traits of empathy, cooperation, attunement and nurture are required for the successful development of children, including male children. Children need warmth, tenderness and love, the same traits that adult males sometimes regard as "being a pussy".
A society cannot function in the long run with these ideas. And since modern ideas and technology now allow women independence, the conflict is no longer avoidable.