Do I need the SQL Express loaded on the management server if we will not be archiving on it?

Yes, you will need it for the management server as well as the archive server. The management server creates the ESM5_2_Central database that holds information such as the root of the device tree, schedules, and reports.
The management server will also hold the aspnetdb database which provides the authentication when installed using internal authentication (as opposed to active directory authentication).

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