No idea if having a rooster around or the lack thereof makes any difference. I have 10 girls, had 12 but one Partridge Chantecler dropped dead (cause unknown) at < 2 years old March last year and an EE was taken by a predator April last year. They have never seen a rooster. If you DO have a rooster, you actually want some of the girls to go broody so you can hatch your own replacements. They don't even have to be the hen that laid the eggs.
Anconas - never have been broody. One has laid 38% (on average of course) of the days since her first egg and the other 40%. Eggs are always Large.
Black Australorps - one does, once or twice a year, the other never has. She is "Super" Echo - best forager, best layer (Large and XLarge), fastest to get back to laying after moult. She has laid 54%, Zorra (XLarge and Jumbo and the occasional

that must have hurt) is only 39%.
Partridge Chanteclers - both broody several times a year. The one I still have is a fabulous layer (Medium eggs, 7 to 10 days in a row then a day off) when she isn't broody but the broody tendency and moult period leave her at 46% of the days since she first laid. The one that died was the same. Still, she is my 3rd best layer by that measure.
Cubalayas - one does (a couple of times a year) the other doesn't. But they aren't called "yard art" for nothing. Even at their best the breed lays 2-3 small (if that big) eggs per week on average. The one that doesn't go broody is at 17%, the other at 22%.
Easter Eggers - never broody. The one that was taken laid at 45%, the remaining one is my second best layer (XLarge and Jumbo) at 51%
Faverolles - Broody as all get out several time a year. They both just got over their first bout of the year. They lay the large end of Small to Medium. One is at 28% the other at 35%.
The point - you can make generalizations about a breed, but each chicken is an individual