Do Employees Leave Managers or Companies?

You've probably heard the cliche "employees leave managers, not companies". As someone invested in helping managers be great, this is important for me to understand.

As I read some of the arguments against the cliche, I see reasons that employees leave such as no development opportunities, poor communication, lack of respect, no recognition of effort, etc.

I agree. However, the manager has the main responsibility for all of those areas (and more). If not, then you don't have managers - you have babysitters or, worse, glorified individual contributors who you are paying too much.