In static excitation system, at starting we can use DC supply or station supply (field flashing). This system has fast response which is essential requirement now. The basic difference between static and brushless excitation is in static system all components are solid state, no moving parts while in brushless system components (diode rectifier) are rotating. Brushless has a merit that there is no brush and slip ring required as in case of static system which led to maintenance problem.
Indeed, you could have batteries to provide field flashing capability for static excitation systems under black-start conditions. But that is something that has to be specified when you are drawing the specs for a new excitation system. Obviously, it is an additional cost, and it also imposes additional maintenance, as compared to the static excitation system without such capabilities.
If the static excitation system is fed from the auxiliary services bus (typically a 4 kV bus), it is conceivable to have an emergency generator to energize that bus for black-start. If the static excitation system is fed from the terminals of the generator, the batteries or an independent DC supply would be required.
Now, note that the permanent magnet generator in a brushless excitation system also requires maintenance. But since the permanent magnet generator (PMG) is needed for normal operation, this maintenance is "easier" to justify and to follow through. In my experience, if there is any break-down in the maintenance in a plant, it starts to show on those emergency equipment. Thus, from the strict point of view of black-start capability, I tend to consider the PMG+brushless excitation system arrangement as more reliable than the static excitation system with external DC supply for field flash. More reliable in the sense of counting on the unit to really start up when required, under black-start conditions.
Finally, note that not all units have to have black-start capability. But in North America, or at least in some regions of North America, generation units are paid for such capability. Thus, at least in my book, these units should be tested and should consistently demonstrate their capability of "delivering the goods" that have been paid for, it is unacceptable for a unit declared as black-start unit (and thus paid for that service) to fail to start up when needed.