"True or false: The only thing keeping men from disengaging in society is other men?"

True.

- -"What can women do to make sure that men engage in society if they just "drop out?"--.

Nothing, if a man is determined to disengage.

In essence you are describing what feminists jealously call "the Patriarchy". That is the arrangement that uses men's cooperative energy to create a civilization complete with technology, and social order.

Of course, men do need to be rewarded for stepping up to the plate and expending this energy. At it's inception, all this meant was a lineage to their children. This is all it took to have a majority of men slaving away, humping brinks and timbers to build cities and roads while the few others designed, invented and created.

Today we see the fruits of this arrangement in our modern world. It allows us to become liberated from each other even as genders go, as is demonstrated by feminism. But the folly of feminism is the insistence on breaking down the system that got us to where we are today. In this regard, the actions of feminism are demotivating to men. And all it would take is one full generation of men to give up on the idea of engaging in the system that disrespects them, and our civilized society will quickly decay into chaos.