Not sure if that's what you mean by soft skills, but I generally think of soft skills as things like ability to communicate well, navigate conflict, and apply emotional intelligence to people management
The "soft skills" as I understand them are more important for roles like team lead/staff, whereas the organizational skills (and maybe even a willingness to be unlikeable at times) are more important for management.
That's why I think a lot of engineers prefer the IC track (I personally think my organizational skills and focus don't qualify me for management, although I enjoy being a lead)
It's interesting, I feel like this applies to a line manager. Once you get beyond that you can just hire to fill in your own weaknesses. If you're bad with organization, just hire a good project manager.
I guess that's how you end up with the Pareto principal. Even just moving from being a line manager to being a manager-of-managers is a different job that requires different skills, but the assumption is if you're a good line manager then the next step is to manage managers. There are probably plenty of people who there who would make fine directors but are bad managers so they don't get that chance. And then you have people who are good managers who end up as bad directors.
I'm sure there's some overlap, but not enough to really warrant the lower role being preparation for the higher role.
A bit like how there are different kinds of programmer positions.
some of this management can be useful but often a lot of it is not, at best. As organizations age companies become mostly about advancing careers of middle management and the things the company actually does become secondary.