Until my kids were in middle school I didn't really give them a whole lot in of input into the decisions we made for them academically. Of my 4 kids the older 3 have been approached and 2 of them have grade skipped.
I have 4 kids and the oldest skipped a couple of grades (will graduate HS at 14) and heading to GA Tech this summer and thanks to dual enrollment will be a junior. He never fought us on the grade skips (he had many) but was disappointed when he was unable to play for the high school baseball team this current year. It is a long story but the short version is I pulled him out of the high school and he became a home school student because the HS wouldn't sign off on the dual enrollment forms for the classes he needed. Nobody argued (school, my son or me) regarding the courses needed but the downstream effect was he wasn't eligible to play baseball. Amazingly despite being young he is a great athlete and did beat out a lot of much older kids last year for his one year at the high school and made JV. Lessons learned here for us - Grade skipping does require sacrifices but the benefits for at least my kid outweighed that (took courses appropriate to him, got 2 years of college done through a double course load + summer school). Socially he has done fine but we keep him in other activities a few hours a week with age appropriate kids. We have no regrets.
My second son is 12yo in 7th grade. He like my older son was course level accelerated. Believe it there are schools where kids are moved fluidly by subjects as well as grade level and this is the norm. In these environments a 3rd grader in 6th grade math or a 5th grader in 7th LA is just normal. You may look for these kind of programs as it limits the concerns your child may have about fitting in. This year my son took mostly 8th, 9th and 10th grade classes and only had 2 7th grade classes. Logistically it was challenging for us moving him between the MS and HS but it worked. Socially he has done fine. The middle school pushed for a grade skip several times but we only recently agreed because we were worried about physical development. This son is on track to be a D1 baseball recruit and we didn't want to harm his chances by him graduating at 16 from HS. After a lot of thought we have been convinced his athletic potential will happen regardless and we shouldn't hold him back academically. He will skip 8th and go to 9th at the HS full time next year, which improves logistics and opens doors for AP classes he couldn't have taken as an 8th grader for college credit. He did not push back because he understood the big picture and we discussed in much detail - pros and cons. We also sought advice from the school district administrators, his travel baseball academy, former MLB players, etc. Who knows if he will make it anywhere with baseball but I wanted to be thoughtful and not reckless with the decision. If he had said no with any reasonable clear argument I would not have pushed him. At 12 we treat him like an adult but would not have been so inclusive with the decision if he was 6.
My 10yo is a 4th grader and we were approached by her magnet public school to skip. She is up a grade in math and science. She was actually excited when we told her but my husband and I decided to decline for now. The reason is there is no rush and she is being allowed subject level acceleration and not being held back academically. She spends much of her day at grade level and some above. Logistically she is able to stay in her school all day and get that. By the time she gets to middle school (7th/8th) she will be further advanced and logistics will get more complicated for us.
We may consider a grade skip at that time because she is tracking to have 1.5yrs of HS credit before she gets to HS. When we told her she wasn't skipping she actually was upset especially since she has seen her older brothers do it. I explained the reasons and she let it go.
My younger son (8) is in the private gifted program where his older siblings started and getting subject level acceleration. He will likely go to 5th grade math next year as a 3rd grader. He is testing now to skip 4th. Many students are fluid up grades at his school so totally normal and the teachers/administrators support and encourage customization as the child needs it.
If your kid isn't really getting what he needs then a grade skip makes a lot of sense. Especially since you say socially he will be fine. My oldest son definitely coasted early in elementary because we hadn't yet found the gifted private school where customization was being done. Just my 2 cents. At 6 the brain isn't developed and the child can't really understand the complexities of a decision like this. As the parent you know what is best and know your kid and what he can handle and how he will succeed. If you have to incentivize so be it. We have pulled those cards too for various things with our kids. Positioning it as this will happen and here are the steps to make the transition go well (e.g. shadowing, review some upper grade material over the summer, etc...) is what I would do. Not yes or no. Going out of your way to allow him to still do fun things with friends his age via sports, after school play dates and so forth may also help. It is hard to move on and leave friends behind but that happens in life all the time. Good luck.
