S1 has a November birthday in a school district that at the time had a 12/31 cutoff. We took the preschool’s advice and redshirted him when it was time for K, even though he was reading at a 3rd grade level at 4.5. As a result, he was one of the oldest in kindergarten. This bonus year gave us the chance to see that his “issues” were not those of immaturity, but that he had sensory integration issues and ADD. As the school’s request, we also got an ed psych eval which was enlightening and confirmed a lot of things we had suspected. As our ped (a HS debating friend of DH’s from Bronx Science) put it, the extra year enabled us to “unstring the spaghetti” and we were able to get appropriate help for him. At the end of K, the school recommended he go straight to second grade. He did, and was 5 yr. 10 mo. old.
It was harder socially when he was younger (noone was interested in the Zoombinis and Redwall), but once he got into some specialized highly gifted programs in fourth grade, things improved tremendously. (Fortunately, he was oblivious to the truly toxic 3rd grade teacher at the neighborhood school.) The HG programs offered subject acceleration and he wound up skipping 4.5 years of math and doing a lot of CS independently (the STEM middle school and high school programs also offered lots of programming, but he was way ahead of the curve there). The best part was that he got to take these classes with his age peers, so we didn’t have to deal with a nine year old going to middle school or early college. The specialized high school STEM program offered complex analysis, lin alg, diff eq, etc. While he could have gone to the nearby flagship for those, he recognized that was a bit out of his comfort zone. Fine with us. His call. He started college at 17 and was fine. Got a math degree and is a software engineer. Is now 25 and still doesn’t drive. 
As an adult, he has told us that in all seriousness, the STEM programs saved his life.
S2 has a Feb. birthday and has always been the tallest guy in the class. Has very good social skills, too, but a lot of LD issues. He was in a 1st/2nd combo class and went with the second graders for reading and math. He also did the HG programs from 4th through HS), but he’s strong across the board vs. his brother, who is very deep in some areas. Skipping a grade came up with S2, but the idea stressed him, and he had a lot of friends and non-classroom interests, so we let it drop and didn’t look back.
When I was in K, the school wanted to skip me at midyear into second grade. (I turned six just after starting K. Was tall and reading at 5th grade level.) Parents said no. At the end of 1st grade, in another school, the school wanted me to skip 2nd and go into 3rd. This time, they said yes. I spent the first quarter in third grade, then the military moved us to another state, where they refused to recognize the skip. Back to 2nd I went. I was always one of the oldest kids and was bored. The teachers used to give me tests to grade (!), independent work and big meaty novels. I spent most of junior high reading history, following Watergate and embroidering during class. In high school I joined everything to keep myself intellectually engaged.
My maternal grandmother skipped two grades and graduated at 15. Instead of being able to accept the scholarship she’d earned, she had to go to work to support the family (1933). She was bitter the rest of her life about that!
tl;dr – It really depends on the kid. One of mine, with the social skills issues) really needed the skip (and more). The other, with similar scores in ed psych testing and great social skills, didn’t need it.