There's no one answer to this. It's highly dependent on the context in which you are working.
For example, I've worked in fields with varying degrees of regulation, where traceability from requirement and change through test case execution is vital. In these cases, having a test case documented in a test case management tool and selecting appropriate test cases for automation tends to be a good solution. Having the test case in the test case management tool also has some advantages for keeping track of the test results as the tools I've used integrate to allow for tracking when the test cases are executed, what they are executed against, the pass/fail status, and attach output (such as screenshots, console output, or log files).
There are several factors to consider, though.
Some tests are "one-off" tests. They may need to be executed very infrequently. If they aren't part of the standard testing suite, they can have a lower priority for automation. However, something else to consider would be the difficulty in setting up and executing the tests - cumbersome tests, even those that are infrequent, may benefit from automation.
There are also different levels of tests. Unit tests, integration tests, and acceptance tests don't all need to be treated the same. The more formality tends to be around system tests and acceptance tests. Still, as your system's criticality or the risks surrounding failures increase, you may want more stringent (and perhaps even bi-directional) traceability around requirements/changes, tests, and code.
You should look at your context. Do you get any benefits by writing tests case up for manual execution before automating? Do those benefits outweigh any costs associated with adding those steps? What about the risks or costs of not writing the tests for manual execution first? Does creating the written test script before automation imply that the test must be run manually, or is it just a way to link a test script with an outside tool? If your system scales, does the process scale (with both increased system complexity and any organizational changes needed to support the more complex system)? Many questions to consider.