What you need to do to break free is simple, but not necessarily easy - get a job with no technical requirements, but which has the potential for technical/cultural solutions.
This is the direction I eventually went in, and one thing I find interesting and notable is that even after solving all sorts of things through programming, people aren't letting me withdraw into just being a programmer - which I think is a good thing. Once I write a program to do something, then I have to document it, then I have to train other people to use it, and then we have to develop contingency plans to do without it if I should be hit by the proverbial bus. And maybe figure out how to get another developer up to speed to maintain it. If I were an actual programmer/developer, then all the different jobs would be pretty much left to other specialists and managers.
Maybe some people would say, just join a startup or make your own, and you will become more well-rounded, but I'm not swamped with offers to do that and have worked in larger organizations.
If your goal is to influence the big picture at a corporation then your title probably has to be Sr. Director, VP, SVP, or something similar. Even then it won't be easy. Political and cultural inertia will conspire to prevent change.