I believe that having a rooster in the flock DOES stimulate the pullets to start laying sooner. A neighbor who lived 2 miles away got EE chicks from the same source on the same day that I did last year. I had roos in my flock (hatched that spring, and sowing their oats freely by the time the pullets were 15 weeks old.) I did give my girls more protein, and they got fermented feed, but they started laying at least 2 weeks earlier than the neighbor's flock. Same thing this year. My pullets started laying 2 weeks earlier than the pullets from 3 other flocks, all 3 of those flocks had chicks that I hatched and gave to them. (The 3rd flock ended up with all roosters from my hatch, which were culled.) So, I want to state that the roosters play a role in the early laying, though it could be also related to the fermented feed or the extra protein. Either way, I consider those early eggs to be a wonderful bonus.
So, to clarify, you don't need a rooster to get eggs, but the rooster may get you eggs earlier.