What sort of life skills SHOULD be taught in schools? And how might teachers go about teaching them?

Make Self improvement books necessary and have some morale class

Active listening and basic counselling skills. I taught these to young people volunteering at a community youth helpline, and they're truly life changing. They changed my life.

How to communicate like a mature person.

If a person has a problem with their coworker, sibling, classmate, parent, etc., the students would be able to communicate in a mature way to mitigate the problem.

The teachers might teach this by showing videos and doing conversation practice.

Real health and exercise classes, financial shit, and sign language. Find the best versions out there that already exist and copy the lesson plan.

Critical thought. Most of the political shit in the world today would never get traction if only critical thought in the masses was a thing.