Getting rid of the lions. Lion males are jerks.
The new matriarchal primates will have a system that is a mix of matrilineal elephants and showy bird males.
Elephants have a totally matriarchal, matrilineal system.
Elephants are a matriarchal society; that is, one that is led by a
head cow, who presides over her herd of females. Each herd is made up
of mothers, daughters, sisters and aunts. They are guided by the
oldest and largest female of the herd. This herd sticks closely
together, rejoicing at the birth of a calf and mourning at the death
of a member.
The herd of females, although maintaining close bonds among
themselves, also interacts well with other herds, families, and clans.
An average herd of immediate family will comprise of 5 to 15 adult
elephants as well as immature males and females...
The male, on the other hand, lives apart from the matriarchal herd,
and travels alone or with other males in a bachelor pod. The drift
from the herd starts during adolescence, at which time the young bulls
start to spend less and less time with the herd. Eventually, the break
is made completely. After this distance is established, the bulls will
live solitary lives, mingling with the females of the family only for
But elephant males are big jerks too, so we will borrow from birds, where the females pick among males who are trying hard to impress them. At mating season, all members of the species get together at the lek. The males show up and compete for mates. They do not fight or bully each other. The males are amazing, colorful and beautiful. They can sing and they can dance and that is what they do. It is the best show ever; something like KCON but with all boys. Reproductive aged females pick the male they like best.
Then the males rest up and drift off, working on their dance moves and songs for next year. Side benefit: the females all get pregnant at once, so they can help each other nurse and raise babies of the same age (borrowed from lions, but it is a very cool thing).
How they look: Females would have a primate body type appropriate for their feeding style, social style and ecosystem. Ground dwelling lemurs and baboons are not so far apart. I would pick baboon-type females because they are durable tough creatures and good for a primate life style.
Male baboons fight a lot and so are robust and have big teeth. Your males are not like that. They do not look much like the females at all. Example: the widowbird.
Males are spectacular Kpop stars with huge eyes, perfect teeth, and shiny manes. They have sweet voices and fluid moves. Only the fittest can keep up such appearances while at the same time finding food and fighting off predators, which is why the birds do it that way. Occasionally a young male might not have his lyrics down, or an older one might be a little more stiff in his dance moves but that is how it goes.