The deployment succeeded, every test passed, and the first real case did nothing. On a Service Cloud go-live, the whole stack of case Flows arrived in production as inactive versions, although each one was active in the sandbox.
This is default behaviour. Active Flows and processes deployed to a production org through change sets or the Metadata API are deployed inactive unless the org has opted in. Sandboxes deploy them active, which is why it never shows up in testing.
Whichever route you take, finish every production deploy by checking that each Flow has the active version you expect, and run one smoke test on a real record. A green deployment tells you the metadata arrived. It does not tell you anything is running.