Hey Hans,
This is a fine form to ask this kind of question :).
I looked into this a bit deeper, and Private IP on Cloud SQL currently
requires the resource to be in the same region. Unfortunately in this case the VPC connector is in the same region with your Serverless deployment, and thus is unable to connect to the Cloud SQL instance via Private IP. There is work being done to remove this limitation, but unfortunately it isn't finished yet.
This is obviously confusing since the docs contradict, but I've filed an internal bug to try and clarify the position on the VPC Serverless page. Sorry for the confusion.
Thanks,
Kurtis