In my experience, yes. Eventually the rooster will be dominant over all hens and should mate with all of them. Even if you don't see it happening, if they run from him he most likely is mating with them. My hens used to run from one of my old roosters too but every so often he would corner them. Roosters are as I would call it 'a bit rapey.'I have 13 hens. 4 EE's. Two lay green eggs and two lay blue. So the green egg layers would be better to breed to? One of my EE's already lays a darker green egg. In between a mint and olive color. But she runs from the roo. Two of the EE's are going on their second year and won't allow the roo to breed them. Will that change as he gets a little older and larger?
If your EE's all come from the same source IE hatchery, they all likely have the same chance of having 2 copies of the blue egg gene. Are they all pea combed? I would personally place preference on the most productive hen's eggs, then egg color darker green being better. That's just me, I prefer productivity and sometimes less productive hens have prettier/darker/more vibrant eggs so you don't want to select for that alone.