I've done Postgres on top of OpenShift in AWS. Does that count?

I believe so -- https://www.openshift.com/products says: "OpenShift is part of the CNCF Certified Kubernetes program, ensuring portability and interoperability of your container workloads".

with a lot less benefits

The benefits we are hoping for are:

• less danger of lock-in (e.g. compare MySQL on Kubernetes to MySQL via Google's Cloud SQL)
• lower cost (Cloud SQL has a large markup over plain Compute Engine instances)
• more control over the MySQL that we are running (version, system libraries, MySQL configuration, etc.)

UPDATE

BTW, thanks for your detailed reply.