Deployment of Temporal

Does this mean that if we were to run ES, that we don’t need to create the visibility database at all?
I know that ES powers some of the search features, but we are not using them. At least not in the foreseeable future.

Yes. No need to create Visibility DB when using ES.

So just keep everything running, and apply the schema changes while Temporal is still running.
Would for example the following upgrade plan be a valid one?

  1. Run version 1.4 (for example) of Temporal.
  2. Temporal 1.5 is released.
  3. Keep Temporal deployed.
  4. Manually start the 1.5 admin tools container.
  5. Run the schema upgrade from the admin container.
  6. If successful, remove the admin tools container.
  7. Upgrade the Temporal Helm chart to start the rolling update.
  8. Done.

This looks correct for Temporal releases with schema changes. Not all Temporal releases have schema updates in which case you can skip the schema upgrade step.
Another thing to remember is never skip releases doing upgrade. Any additional steps needed to upgrade are carefully documented as part of release notes. So please go over them carefully.