I have a few already deployed and running functions that use a custom VPC connector (projects/dc-****/locations/us-east1/connectors/serverless-vpc-dev-use), then I created a Cloud SQL from scratch with only the public IP.
I've been able to connect to this Cloud SQL Instance using the cloud_sql_proxy, but I could not connect to it from my Cloud Functions using the same Cloud SQL public IP.
What am I missing?
It's impossible to tell what you are doing wrong because you've provided no details -- things like what your connect code looks like, what error you are getting, and what you've tried so far can help determine what you are doing wrong.
If you are using public IP, the VPC connector shouldn't matter. Try following the directions in the documentation on how to connect here: https://cloud.google.com/sql/docs/mysql/connect-functions
