Both my kids are late born.
At my kids public school, Kindergarten is for 5 year olds. Per the cut-off dates, per my district for public school, both my kids started Kindergarten at 4 then turned 5. They made the cut-off dates. I have a son and daughter. Both genders. They both entered into Kindergarten at those ages, and they were FINE.
Both also had preschool. And they did fine.
It is not just about age.
But also about the child, themselves.
I know many kids, that were held back. And entered Kinder at 5 then turned 6. They were fine. But most of the kids that entered Kinder when my kids did, were also late born. And they did fine. Many of the students in my kids' classes, are their same age. And it was the minority, that was older.
For those that entered at a later age, sure, they are older. Some of them also re-did Kinder, their parents held them back, and so they repeated Kindergarten. But those kids (I work at the school), are physically more "mature", they are taller/bigger, and look, older, than their classmates.
But, it does not necessarily, make a kid better than the others (both academically nor emotionally), than the kids that started Kinder earlier.
I see it, myself. Many of the kids that did start at an older age and/or was retained in Kindergarten, are not that much improved, emotionally or per their maturity. I work at the school, and I see it myself. And quite frankly, those kids that are "older" and/or held back and repeated Kindergarten for whatever reason, their is no "improvement" in them, nor in maturity or emotionally or in behavior or academics. They just look and are, older. And some even will, regress and/or still act like they are younger than they are, even if they are the oldest in the class. So they look a bit out of place. Because, physically, they are more mature.
You go according to your individual child.
And per the cut-off dates, of the school.
Keep in mind, that at the school my kids are at, IF a parent, entered their child into "school" at 6 years old, they are put into, 1st grade. UNLESS the parent, specifies, that they want their child in Kindergarten.
So, you NEED to, find out from your kids' school, what their rules are.
It is not the same in each State or in each school.