I think your concern is shared by many parents of GT kids. It's hard to know if they are appropriately challenged or if there are gaps. I would say that many if not most GT kids will be fine with gaps. In fact, they are kind if nice because it means there is something that teachers can do--teach a new concept. A problem that often occurs, however, is that many teachers see a gap as "not ready for this level" when in fact most GT kids will get the new concept just fine with a short lesson.
One way to find out if your kid is ready for a higher level if instruction is to use pretests -- unit or end of grade. If you child shows mastery on 85% of things he has not already been taught, then maybe he should move to the next level.
You mention that GT programming does no start until 3rd grade. Our local school district was the same, but we reached out to the GT coordinator and asked doe advice and help as it seemed that our son needed higher level math and reading. She worke with the teacher and principal to get our kiddo tested and differentiate during kindergarten, then then he skipped 1st grade. The school wa very flexible and worked to differentiate afte the skip and before official GT programming started in 3rd. We found pace to still be an issue and transferred our son to a school for HG kids that has acceleeated programming, but if you don't have FT GT schools nearby, you can try to work with your local schools.
Our son had similar issues with handwriting, and that proved to be a big issue at our son's local school in terms of getting into appropriate classes with other kids. My DS was ready for higher level math, but the school wouldn't let him go to a class with older kids because they said he couldn't handle the writing. Luckily the teachers at the new school didn't see the as a problem if DS could explain somehow (eg verbally) that he understood the concepts. Be prepared for a lot of advocacy!
