jack'smom,
The problem isn't that I "need to relax and give [my son] time to develop.", but that I need to run to keep up with him and his crazy development.
No need to prickle; Jack'sMom's "you need to relax" isn't the "you need to relax" you hear from other people ;-) She's right: however much you know your DS is gifted, you have
no idea right now whether a 3-5 year grade skip is going to be appropriate for him. (Based on what I read and hear here and elsewhere, it's vanishingly rare for it to be a good answer, actually; you don't solve what is basically a pace problem by starting at a different point.)
I too had a miserable school experience in which I wasn't challenged, so I know what you mean about being determined not to let that happen to your child. I'm not advocating doing nothing.
You say your son is (about to be?) doing handwriting at K level, and you posted elsewhere about how he doesn't currently like teacher-structured environments. Those are the two really big things that limit the practicality of large skips: they have to be able to meet the expectations of the class they go into and they have to be able to produce the output. (Some schools will be prepared to do some accommodation e.g. allowing typed rather than handwritten output, but it's not a silver bullet: my DS still doesn't find typing any easier than handwriting.)
My DS7 is, as I often feel obliged to put in, untested, but on the pure intellectual side, a 5 year skip would look just fine, in the sense that he can easily handle explanations etc. aimed at 12yos, make arguments expected of them, etc. In fact he'd still need individual teaching in his strength areas. However, although we've discussed skips, we've decided against even one because it isn't, overall, in his best interests to do it. The output and social expectations are an important part of the reason: e.g. I'd rather have him somewhere where he can, with effort, meet output expectations than somewhere where he has to have accommodations because it's obvious he can't meet them. Another major reason is that no amount of skip would leave him looking typical for the class he arrived in. There's no substitute for sensitive teaching, and if you've got sensitive teaching, it can happen in a variety of settings.
Concretely, at 7 DS is currently expected (ha :-( ) to be able to produce a nearly full A4 page of writing (small, neat, joined up) on a set topic on demand in about 45 minutes. That may, I think, be a little ahead of usual expectation for that age, but it's a guide. He has to be able to do what he's told all day, get himself to a lesson on the other side of the campus on time and on his own, dress himself quickly and without fuss including tie and lace up shoes, etc. etc. etc.
TBH I think if you try to grease wheels for a 3-5 year skip for a child currently only just three years old you're simply going to get backs up and get yourself written off as crazy. Keep quiet, homeschool him for a bit if it seems right, see how it goes, and if you want to put him in school several years ahead of age in due course, then get a copy of the Iowa Acceleration Manual, get an IQ test, and go for it. But argue it then on the basis of the evidence you have then that it's the right thing. For now do what's right for him now.