This is all very good advice, but I would caution anyone asking the question at the start of the essay that they probably shouldn't be a Staff Eng. Unless you're at a company where that title is simply a rung on the ladder that doesn't have differentiated responsibilities (there are lots of those out there).

Every person I've worked with who's been successful as a Staff+ Engineer, promoting them was normally more of a formality since they were already clearly doing the work. Every person I've seen 'rise to their level of incompetence' was striving to get the title/pay bump and looking to 'play the game' to get there.

If your motivation for solving people's problems is that it will get you a promotion, instead of the fact that you like solving problems, then it's probably not the right job for you.