What are the advantages of using the concept of a lever arm when working with torques?

I tutor some high school students, mostly in math but occasionally in Physics as well; I love the science but only took it through the high school level plus a couple of first-year university courses. I’ve noticed that nowadays almost all texts use the concept of a lever arm when introducing the idea of torque. This was an idea that I only ran across when I started tutoring; when I learned about torque in high school (I can’t remember whether it was part of my university courses), we were simply taught to use the component of the force perpendicular to the moment arm, an idea that immediately made sense to me. I understand that computationally the two approaches are equivalent, but it seems to me (as a non-Physicist at least) that the use of a lever arm is a much less intuitive approach than the way I was taught. I would love to get some idea of why this approach is now preferred.