Kdg. teacher here. Find out what the 1st grade teacher's expectations for her students are. If they're expected to read, then you will have something solid to base your decision on.
Does your DD know her letter sounds? In order to read, you have to be able to recognize your alphabet letters randomly (not rote - A, B, C). She also needs to know the sounds the letters make including the short vowel sounds. Is she able to recognize letter sounds in words? For example, can she pick out the b, the u and the g in the word bug? These are the things she needs to do before she can learn to read.
From there, you can work on sounding out the letters and blending them together (ie: start w/easy endings -at, -ot, -an, -ap, -et, -up, etc.). Get her to blend those sounds and then add the beginning letter sound (ie: c-a-t) and have her sound and blend those letter sounds to form words. Also, sound the letters out (ie: make the b sound, the u sound and the g sound)and ask her to blend those sounds together out loud and ask her what word you're trying to say.
Is her school sending home sight words for her to learn? A Kdg. student should recognize 100 sight words before going into 1st grade. You can find these on the internet. These are words your child needs to know by looking at them rather than sounding them out (ie: me, he, she, the, my, that, and, etc.).
Right now, my students are sounding out words and reading them. They are also taking home a "sight word journal" which is a tablet containing 10 words for them to learn each week. I started off w/5 words a month ago and built up to 10. Note -- not every word is a sight word -- about half of the 10 words are words they can sound out. Parents are expected to help their child learn these words. We use them every day in class, so the journals are extra practice.
I am not sure about your child's program at school ... they may do more beginning to read in 1st grade. I know that, in my school, it is my job to have them sounding out and blending sounds into words and actually reading small sentences (The bug is in the rug.). However, every school and school system differs. You could always ask the 1st grade teacher what she expects of your child.
As for holding a child back ... I find it preferential to hold a child back in Kdg. over an upper level. If you hold back in the upper level, I find that's when the "failure chain" starts -- they think they're stupid, they wonder why they should bother, they feel like failures.
Why don't you also talk to the Kdg. teacher about why she's not started to read. Maybe there are some good answers.
As for my students, we just started our unit on Geometry (solid shapes - cone, sphere, cube, cylinder and fractions) and I'm worried that they're not going to get to the addition and subtraction portion of math before the end of the year. I have, through various other exercises, taught some addition and subtraction ... but I'd prefer if they had some more of it. I know that the 1st grade teacher will start where I left off, so if I don't cover the entire unit of addition/subtraction, I know she will start off w/that. This year, the students have a good grasp on the Reading/LA/Phonics and a not-so-great grasp on the math. Some units have taken longer for them to understand, so we're working at a bit of a slower math pace w/this group.
Good luck to you ... you can always work on teaching her to read over the summer. But, remember, in order to read, she needs to know that alphabet randomly, as well as the letter sounds!
***ETA -- Everything I've stated goes on what my school expects of its students and of me, as their teacher. I am not saying that every kid in the USA who is in Kdg. should know their 100 sight words or s/b ready to read going into 1st grade -- it's just what my school expects of its students!!!! ***