What could I be missing with Prisma client, Cloud Run, and Cloud SQL - my Prisma client can't socket-connect to my Cloud SQL instance DB? - sockets

Question

Background
I have a NestJS project with Prisma ORM, and I am continually receiving the error:
PrismaClientInitializationError: Can't reach database server at `localhost`:`5432`
This is happening during the Cloud Build Deploy step.
Since this is a containerized application (attempting to) run in a Cloud Run instance, I'm supposed to use a socket connection. Here's the documentation from Prisma on connecting to a Postgres DB through a socket connection: https://www.prisma.io/docs/concepts/database-connectors/postgresql#connecting-via-sockets
Connecting via sockets
To connect to your PostgreSQL database via sockets, you must add a host field as a query parameter to the connection URL (instead of setting it as the host part of the URI). The value of this parameter then must point to the directory that contains the socket, e.g.: postgresql://USER:PASSWORD#localhost/database?host=/var/run/postgresql/
Note that localhost is required, the value itself is ignored and can be anything.
I've done this to the letter, as described in the Cloud SQL documentation, with the exception that I percent-encoded my path to the directory containing the socket. I've included and excluded the trailing slash.
So my host var looks like this, mapped from the percent-encoded values:
/cloudsql/<MY CLOUD SQL CONNECTION NAME>/<DB>
I've read over the Cloud Run documentation, and in my mind, I should expect a different error if the instance itself can't connect to the Cloud SQL instance. I've followed the "Make sure you have the appropriate permissions and connection" from the documentation a few times now.
Is there anything obvious that I'm missing? Am I wrong about an error related to Cloud Run instance just not connecting with Cloud SQL instance?
Things I've tried & things I know
I CAN connect directly to the Cloud SQL instance locally through psql
I CAN run a local server with the Cloud SQL instance public IP and establish a client connection & interact with the database
I CAN successfully create an image and run a container from that image locally
My big concern
It doesn't make sense to me in which order things should connect to the Cloud SQL instance. To me, the Cloud Run - Cloud SQL connection MUST be established before the application run inside the Cloud Run instance can establish its connection through the socket to the Cloud SQL instance. -- Am I thinking through that correctly?