Hmmm, that's a good guess, but can't they just run an industrial diesel generator prior to start up?
When I was reading about the threat of an EMP I read that it would take weeks for power plants to go back online.
They could, but the electricity demands can be substantial.
A large coal power plant (say 2000 MW) can have an internal electrical load of about 100 MW. That's way beyond diesel generator size, and is a respectable sized power station in itself.
Remember, you've got conveyors moving many tonnes of coal a minute lifting it up large heights. Coal crushers which grind thousands of pounds of coal a second to a fine dust. 10,000 horsepower water pumps to supply water to the boilers, heaters to get the boiler up to temperature so that the coal can ignite, 10,000 hp compressors for pressurising the combustion chamber, etc.
A nuke plant has several diesel generators, but these only power the bare minimum safety systems, yet they typically use 10,000 hp gensets, which are just about enough to run the reactor emergency cooling pumps, rather than be able to power all operational systems.
It then comes down to cost. Is it necessary to have so much diesel generation on site, just to be able to "black start" a power station? In general, it's not. If you've got a grid supply, it's much easier to use this, and only maintain "black start" capability at a few select plants (often those where the startup demands are minimal - hydro and nat gas), and then use a sequenced start.
A battery is used to start a diesel genset, which is able to operate the hydraulics on a hydro plant. The water turns the turbine and starts the main generator, Once up to speed, the generator is connected to the grid. A nearby nat gas plant connects to the grid and using grid power it starts its turbines, and synchronises the generator with the grid. Once connected, a coal power station can start its conveyors, crushers and compressors and start warming its boiler. Once up to temperature it starts its turbines and synchronises with the grid. Now the grid is strong enough to start spooling up lots more power plants, and start reconnecting loads.