Pedigree is a necessary but insufficient aspect. Pedigree is a marketing instrument, but you need to actually market yourself to stand out. Plenty of ivy people unsuccessfully set up their practice, and people who are not ivys can succeed (albeit with more barriers), and depends on where you are. In the Midwest, you generally don't need ivy, but something name brand is *still* helpful.
Realizing that full-on cash subspecialty psych is one of the cushiest gigs in all of medicine, it's probably best for you to plan to gun as much as possible during training and pretend that you are shooting for derm or retinal surgery, if this is what you want to do with your life.