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 :-(

Not in my experience. I think a key thing is that a principal engineer is technical glue. You run around talking people getting insight on what is happening on the front-line, and then you nudge things in directions such that the game wins.

Does this meaning providing tactical insight for the project? yes.

Does this mean questioning priorities and aligning priorities? yes.

Does this mean that making sure the department is sustainable on hiring objectives? yes.

You could say that the role is being a smart technical person that is responsible at the level of "almost stake holder". A key thing here is that a principal engineer should also have significant stock. At a big company, this should be in the seven figures.