This does vary based on the field and kind of school. It's definitely accurate for being a professor at a top-tier research university in the sciences or engineering. A prof job at those kinds of places could accurately be titled "research manager": the main job requirements are managing and bringing in funding for a fairly large research lab of 5-10 grad students and possibly some research scientists and postdocs.

But you can shift the balance of the various components if you go to different kinds of schools. To take the opposite end of the spectrum, if you're the CS prof at a small liberal-arts college, your job will involve a lot more teaching and mentoring undergrads, and a lot less grant-writing and research-lab management. It's unfortunately not true that there's a happy medium to fit every kind of temperament, but there are some options at least.