Hi. We had the same problem. Our ds was in a small private Catholic school, and the talking became a major problem. My ds6 has been tested, and even with the test results, this particular school was unable to provide a differentiated curriculum or advance him to first grade. That said, it's not a public school, and you would probably have more luck there.
We determined that our ds was bored, and he, too, was quite put off by the fact that he had to color and do "baby work" all day. He said he spent his whole day waiting on everyone else to finish their work. His words, not mine. The school was concerned about adhd, and honestly, I'm still not convinced that he doesn't have a mild case of this. BUT, I do know that when he is sufficiently challenged and interested, the impulsivity and inattention are nonexistent. I find it hard to keep him challenged and engaged myself. He's still fidgety, but both his father and I are that way. No way he could escape it! : ) Your school seems to be much more supportive; maybe the teacher could provide a differentiated curriculum to keep him challenged and interested.
If you google "adhd vs. gifted" you'll get lots of good information on the similarities in behavior between kids with adhd and gifted children who are bored and unchallenged. We ended up pulling ds out of school and homeschooling because I wanted time to find the right placement/school for my son. Unfortunately, our public schools are not very good, and the negative influence there is pervasive. So, we'll see what the future holds.
Anyway, I think what you are describing is discussed quite often on this board. I think you are in the right place and thinking along the right lines. As for the testing, I wanted to know for sure; I'm a numbers kind of person so having the results helped me not to question the giftedness. Good luck, and I hope this was some help.