How Much of a Shortcut Are Connections in Top AI Lab Hiring for PhD grads? [D]

Hi everyone. I'm trying to calibrate my expectations and would appreciate honest perspectives from people involved/ with experience in hiring at places like Anthropic, OpenAI, Google DeepMind, Meta, etc (haven't started interviewing yet). I'm at a top ML university, but my advisor is not particularly well known in industry and doesn't have many industry connections. Looking around, I'm seeing peers with research records that seem comparable to mine (and in some cases arguably weaker) land interviews and jobs at top labs.