I would say that’s not entirely true. Your faculty advisor is very important, but your department is too, and how much they are each important varies depending on the kind of PhD. For the sciences and some social sciences, advisor/PI might be more important because they have much more control and direction over your program; for the humanities fields (and some social sciences), you work more independently and your department’s strength might be more important than the one person you work with. Besides, you don’t want to be with a strong advisor in a weak department, because that will also affect the coursework you take, your peers, and the resources available to you.
And yes, I would say that my field and most other social and natural sciences expect you to have a specific area of study in mind. That doesn’t mean that you can’t change it (slightly) over the course of your PhD, but you do need to have a general broad area picked out. I’m not sure about computer science, though.