I've been helping other people through the Google hiring process, as I'm about to graduate from my PhD and head off to Google (I went via the intern conversion, so my experience is atypical). I know a couple of talented undergrads, and some have applied there.
The recruiters have always asked every one that I knew of "who do you know at Google?" I know this because they have come to me saying "The recruiter is asking who I know and you're the only person, can I say you?" They all seem to have a sense of urgency in the question, which indicates to me they've been told it's important.
I think that the OP made it to phone screen alone indicates that he was past the first cut. I would be surprised if referrals matter from that point on, as you say.
The recruiters have always asked every one that I knew of "who do you know at Google?" I know this because they have come to me saying "The recruiter is asking who I know and you're the only person, can I say you?" They all seem to have a sense of urgency in the question, which indicates to me they've been told it's important.
I think that the OP made it to phone screen alone indicates that he was past the first cut. I would be surprised if referrals matter from that point on, as you say.