K.L.
II think you have the wrong idea here of what the purpose of kindergarten is - and that purpose has NOTHING to do with academics, and appropriately so.
I am a teacher - I started in regular education in public schools and have since moved onto special education, psychology, montessori based curriculum, and generally teach in the private realm. However, I was a kindergarten teacher for many years for the public school system and the benefit of kindergarten for ALL children does not lie in what is academically taught - nowadays, most native english speakers already know beyond what is taught academically in kindergarten (alphabet, colors, shapes, counting past 10, etc) and many of them come into kindergarten with pre-reading skills. The benefit lies in those basic skills that can ONLY be learned in a classroom setting that will be the basis of their school lives until adulthood: learning to work with others, helping each other, listening, attention and focus skills, taking turns, social appropriateness of behavior.... I could go on and on. Not all children have the benefit of siblings, and for many, this is their first experiences in a large social situation with children from all socioeconomic statuses and backgrounds accounted for. In the Bay Area, the diversity of culture in the classroom is very wide, and it is a truly enlightening thing when children learn about the wealth of meaning and experience of other cultures. No, kindergarten is NOT what you learn about letters and numbers, but where children many begin to learn empathy and understanding, and about how to work together. They will also learn how to stand up for themselves and about how to react in situations where they may feel uncomfortable or different - they will learn to respect authority (we hope) and perhaps build a better sense of right and wrong. There is so so much more to it than just academics. It is NEVER recommended that a child skip kindergarten, and I would strongly suggest against any private school who's kindergarten program is academically based and sends their children home with any sort of real quantifiable homework.
In the pubic school system, children are generally tested for giftedness at the end of second grade/beginning of third grade if the teachers feel it is appropriate. If the child seems like the right sort of child for the GATE or accelerated program, they will be placed in a classroom that is either a split grade class or just one grade, taught by a "gifted and talented education" GATE educated teacher. In a private school setting, things can be entirely different because private schools can pretty much conduct their teaching in the way they wish. Private school curriculum is often accelerated, so if you want to place your child in a situation where you will be paying tuition for an accelerated program, choose one of those private montessori or charter schools (I have worked in both, and I do not necessarily recommend them for all children. They are not for most children).
I was a very gifted child and was placed in gifted and talented classrooms and actually BUSSED once or twice a week to another school while in elementary to take part in their special gate program. WHile somewhat singled out because everyone was aware I got to leave once a week, I still had very normal school friends, did normal school work, and had a really great typical school experience in which I was allowed to be a child. I do find the public school GATE programs to be better than those in private schools as it still allows those talented children to have somewhat of a normal school experience while still being enriched. Children educated only in the private school system tend to have a very different experience of what school is and can really have social issues when interacting with the "outside" world - which is the world they will be expected to join once old enough to join the workforce.
Be happy that your child is gifted! You are the best person to foster a love of reading, learning and academics with her at home. Go on interesting outings, visit museums, experience things that might seem a little bit mature for her age and see how she does. ENCOURAGE HER TO READ, because reading really is THE basis for the enrichment of the mind --- let her attend kindergarten, and worry about the academics later. Allow her to be a child before throwing her into a situation where there is pressure to perform or succeed academically - if she is a gifted child, chances are, she wont need anyone to push her to do her best and learn.
My 2 year old is so smart it astonishes me (between myself, stanford educated, and my husband that works at Apple, she got some GREAT smart genes) - be proud of it! but the most important thing to do is never get a big head about your child's abilities - other parents will see right through it and resent you. My 27 month old is reading very simple words and I will always work with her and encourage her reading skills, but she will go to kindergarten and attend public school just like I did and be gate identified when the time comes. I will enrich her life in other ways with piano lessons once she turns 4, dance if she wants, or any other class she wants to take. It is our job to give our children what they need if school is too easy for them academically - school provides a wealth of other skills that are JUST as important.