The last person he nominated was a Republican back in May 2022. He was not required to nominate a Republican. The board can only have a maximum of five governors from the same party. Currently there are three Democrats, three Republicans, and one independent and also two vacant spots. So there could easily be two more Democrats. If the last person he nominated was a Democrat, that would mean four Democrats, two Republicans, and one independent which means it would not matter if that independent person always sided with Republicans.
So he essentially had the power to get DeJoy removed for nearly the last two years.
The source for the info is Wikipedia which lists the when the terms began for each governor.