The same thing that happens to other accounts with no named beneficiary.
I don't get mailed statements for ANY of my accounts.
If you're too dumb to create "if I die" documentation and keep it current and let people who need to know about it know about it, too bad for you!
Adults need to take some responsibility for themselves. It's not asking too much.
Personally, I called GS and named my wife as beneficiary within moments of opening the account. I updated my "if I die" file shortly thereafter.
Or maybe, maybe we can call this a miss? It's a common feature, they really should have it.