I think privilege is important, but not in the immediate sense. If you are using your connections to get the same job you might have gotten by padding your resume or memorizing the tech interview quiz, that's not going around the corner, that's more like getting pulled through the little door. You're still in the same place. In a sense, all of the networking strategies are just another game, the same as interview strategies. It's just that that's a game that isn't accessible to many people with less privilege.
If I remember correctly, SpaceX has a spot on their job application for something like "show us something cool that you've worked on in the past." That's trying to get at the idea of "going around the corner" and finding people who are creative, who candemostrate that they can function in an environment where there isn't a set of instructions to follow. That's not something that you get to do if you're barely surviving. There's privilege in having had the opportunity to be creative in the past, and those who have had more opportunities are in many ways more valuable to companies than those who have not had the same opportunities. The problem is that it's virtually impossible to distinguish between people who have never had opportunities and those who just never took the opportunities that were presented to them.