I have somewhat strong feelings about it. Both my sons could have skipped kindergarten and one both kindergarten and first grade. I opted to keep them with their peers. We supplement at home and would continue to do so so I did not see the point of skipping them.
One of my sons who is now in second grade was doing second grade work in kindergarten. Instead of skipping him, they allowed him to go to the second grade room only for reading and math and the rest of his day was spent in kindergarten. He was bullied, called a stupid baby, stopped wanting to go to school and at one point actually pooped his pants. I said absolutely not, no way.
My daughter on the other hand has a late birthday. We started her on time which seemed early since most of the kids would turn 6 in kindergarten. She has struggled to fit in socially as she has always been less mature than her classmates, hit puberty later, etc. Sports really saved her.
For my sons, it's still a struggle finding challenging work for them but we just learned about a program called Highly Capable Learners for which he's been nominated. The school district provides curriculum that he can do in his classroom with his peers. I think that is great.
Just because a kid reads well or is great at math doesn't mean they are cognitively on the same level as kids who work on their level. The class discussions are really helpful and age-appropriate.
I volunteer in all my kids classes and see both the pros and cons to skipping. They fit in so well but get bored easily with the work