Just how important is university prestige when completing PhD?
I guess that if you want to teach at Princeton (or Oxford) then go to a place just like Princeton (or Oxford). Otherwise, I think that what you do and the work you complete will have a much bigger impact on your future than the prestige of your institution, which may be overrated. There are a number of R1/Ivy graduates in the news that I consider to have the intelligence of rocks.

Having an advisor with lots of ideas, wherever they are, is a big advantage. Being able to work closely with your advisor is a big advantage in doing good work, provided the advisor does good work. 

And, your future may be determined as much by economic conditions as anything. You get no guarantees. In fact, even if you go to Princeton, expecting to teach there later, there is no guarantee that any suitable job will be available there (or at a similar place) at the time you need it. 

Choose your institution on a more sophisticated set of criteria than it's perceived reputation. 

I went to a big place with a lot of faculty and grad students. But the number of people doing anything that I was seriously interested in was very small. Three or four faculty and half a dozen students. 



Actually, I should note that many universities (not all) have a policy of not hiring their own graduates. This is to make it more possible to bring in new ideas from the outside, as well as spread their own reputation more widely. Some will make exceptions, but only for the most exceptional candidate. As an aside, I recommended that my own BS and MS students go elsewhere for their next degree, just to be able to bump up against fresh ways of thinking.