But the "grid power" needed to run the reactor's control systems, etc., can almost surely be generated by the reactor itself while it's online, right?
A naval nuclear reactor needs external power only to get started; when the ship is in port, this will typically be shore power from the grid, and either in port or at sea the external power could come from the onboard emergency diesel generators. When the reactor is started up, it will eventually "go critical," i.e., achieve self-sustaining fission. After that happens, one or more in-plant, steam-turbine electrical generators will be started and brought online to provide electrical power for the ship, including the reactor control systems, etc. [0] A similar arrangement seems likely to be used for civilian power plants. [1]
[0] e.g., https://fas.org/man/dod-101/sys/ship/eng/reactor.html
[1] e.g., https://www.elprocus.com/what-is-a-nuclear-power-plant-worki...