When my ds10 was 5 yrs old, he was in pre-k due to a late birthday and whizzed through to 2nd grade curriculum within a few months. He was starting to do 3rd grade work until the school (which was a gifted private one) said that they could no longer accommodate him; or wouldn't accommodate him - even though this school went through to high school.
This school was more like a traditional, structured gifted school and in many ways ds did well there. Unfortunately, though, as I mentioned, the school refused to let my son attend after he started to do 3rd grade work as a 5-yr-old and bump him up. Hopefully, this won't be the case for you though. My son did attend a Montessori type, unstructured gifted school for a little over a year afterward before we started to homeschool him for 1st grade, at 6.5-years-old.
So I'd say to let your ds go, but to try to supplement at home and watch out closely for boredom. It's quite possible that your son is bored at school and doesn't offer any challenges or ways for him to be intellectually curious. If so, you may try to work with the school to come up with some kind of strategy to stem you ds from getting bored - provided they're willing.
Yes, I agree with you on music and having start piano if you can. A foreign language/s might help to slow your ds down. It's helped with mine in many ways, including dealing with the perfectionism. Also, try getting him to type or do as many 'extra' things to engage that busy, active mind of his. Try art, different sports, or anything else where there is a spark of interest, I say.
Swimming is an excellent activity regardless because very few of us will ever actually swim competitively (or ever want to be competitive swimmers) and it works on all the core body muscles. Martial arts may help too.
Also, your ds may seem to be more external motivated, but I wonder if it's the case that he knows how different the other kids are from him and can't help himself. Plus, with an older brother, he may want to appear different from him. I know with our ds, he had a lot of trouble relating to the other students because he was just so far out from the other kids in school and this just escalated the more he rapidly accelerated.
Since my ds is a 2e and probably more like your older son, he tends to withdraw and hide his pgness but at times he used to get really annoyed and frustrated with the snail pace of school and the other students and then, of course, blow up.
