Not in my experience: there are typically specific project and program managers. The job of the Staff/Principal Engineer is usually to perform some kind of tech

Not in my experience: there are typically specific project and program managers. The job of the Staff/Principal Engineer is usually to perform some kind of technical leadership: guiding code reviews, mentoring, architectural or technology guidance (what framework should use use?), remember the lessons learned from previous projects, and also serve as a liaison between the technical and non-technical departments in a company. And in most cases, that's in addition to writing code. One of my last job descriptions as a Staff Engineer had me writing code about 10% of the time :-(