Yes: IIRC some large generators work exactly like this, as the energized rotor gives a lot more flexibility in managing frequency and power output.

Not just some, approximately all of them. It greatly complicates the logistics of a black start. † Of course that situation has additional complexity due to the need for substantial additional power in order for the various fuel supply systems to operate but I digress.

† https://en.wikipedia.org/wiki/Black_start