A promotion doesn't mean you have to take on the role of your manager, it generally means you have expanded (or new) responsibilities and expectations. For larger companies career progression broadly falls under either an individual contributor (IC) track or a leadership track. As an IC you can grow your career from Engineer to Senior Engineer to Staff Engineer without changing managers or teams, and there is a similar progression for leadership.

Becoming a manager because you think that is the only ladder to climb is definitely the wrong choice. I think a great manager is also a leader who takes an active interest in the motivation, growth, and outcomes of their team. If a management team worked the way you describe (manager's managers run the show and broker deals with their peers for micromanaged promotions), that would be an extremely dysfunctional environment.

In my experience it is always the direct manager who recommends someone for promotion rather than the next level up (although your manager's manager likely has final approval on budget). As a manager, I support and coach my team toward their career growth aspirations with regular performance & growth conversations. I also try to ensure my direct manager has visibility on the state of the team, but ultimately they are expected to be looking at a bigger picture and trust that I am taking care of my team.

They're both important if you are in a large corp.

They both need to praise and recommend you in order for you to be promoted (your manager has more to do on that). A bad word from either will kill any prospect of promotion instantly.

They both need to carry their weight for the team and the department respectively. They put a ceiling on what your team and your department can do and can get (work, budget, opportunities). In a bad case there's simply no slot for promotion so you can't be promoted. In the worst case your team/department might get gutted.