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“something like 3kw to energize the magnetic field to produce electricity - just like those old car generators.”

This is inherent to all synchronous generators. To energize the field coils of a nuclear power plants main steam turbine generators is measured in the megawatts it’s usually 100 to 1 for every one hundred megawatts of output you need one megawatt worth of field currents. This energy is first provided by an external source once the turbine is running it is backfed from the output of the turbine itself usually as a separate DC generator on the same shaft to avoid a harmonic feedback issue. Wind turbines do the same thing they are energized by external means until they are spun up then like any other synchronous generator become self energizing.

This is why black start capability is only found in a few thermal plants just because the main turbines are spinning via nuclear power or gas turbines without the grid to synchronize too and draw their field coil energization from those turbines are useless.

For black start a separate on site Dynamo must be present usually a permanent magnet DC generator driven by a fairly large diesel or gas turbine. That is spun up first since it’s permanent magnet DC generator it immediately puts out HVDC current that is feed to the field coils of a single main turbine beginning it’s AC generating cycle then it is spun up to synchronous speed to output 60Hz AC. This becomes the seed generator the others on site are then feed HVAC current from that generator into their field coils bringing them into synchronous not only with the field current but also to each other. At that point This plant exports power to special dedicated lines to a near by plant it top spins up and uses those dedicated lines to energize its field coils in sync with the other generators already running. This process continues until the whole grid is back up it can take weeks.

Texas came within 3 minutes of having to do a black start when the natural gas grid lost 65% of its capacity during the freeze. We in the industry warned them in 2008,2011,2012 to winterize the well heads and surface lines plus gas processing plants we were ignored.