I know you asked about girls specifically, but I'll give you both ends of the spectrum.
Our son's birthday is August 29th. The cut off was August 31st. We started him, so technically he was only 4 for the first couple of weeks in kindergarten. I've posted this before when this question has come up, because the ONLY time it was an issue was in HS when all the buds were driving, sometimes an entire year before he was! He did fine in school and sports and had (still has) a ton of friends.
Our daughter's birthday is October 1st. There are 4 years between the kids, but 5 years in school because she missed the cut off and had to wait another year. She wanted 'homework' too when her brother started coming home with spelling words, etc. So we would find things for her to do and we had her in a pretty progressive pre-school. I remember calling the district we lived in at the time if she could take some kind of assessment test to go ahead and start her in kindergarten the year she turned 5 and was told NO! She wasn't 5 by the cut off, period, end of discussion. So she was more than prepared an entire year later, and has continued to stay on top of things at school.
*Sorry, if you allow me this moment of digression. . . Proud Mom declaration coming up. She just finished her junior year in a private, college prep school. With the exception of a B in Physics, she had all A's! She had English and American History, Honors Business Finance and Calculus, Advanced Spanish Topics and French 2.
Now, her personality has always been kinda focused and driven. So I can't she did so well, because she's the oldest in her class. LIke I said, this is a different school environment, so most kids are pretty focused.
Socially. . . again, she has done fine.
My feeling, and from my experience, kids will be just fine.