There are two types of children whose parents come to me for chess lessons: cerebral kids who have possibly already tried chess and have already beaten everyone in the household, and rambunctious, smart kids who need to learn focus. Sometimes, the same child has both characteristics.
Can learning chess change the very nature of a child? The answer is: No, not in any way that you would likely notice in the short-term. Learning chess is a wonderful tool for the tool box. He or she will use it as they will, either now or much later. And of course, most children, no matter how smart they are, wish to spend some of their day pursuing less obviously cerebral activities since they are children and that is their nature. Remaining fit and preparing to be an adult requires a certain amount of physical engagement. If you are a kitten, you play-hunt. Nature tells you to do that just in case you will not be a house cat all your life.
If a child is restless despite karate and little league, and this restlessness seems to go beyond normal childhood parameters, examine the physical and environmental factors that may be the cause. If the issue is that they are not having their intellect challenged enough, then yes, chess can help. If there is another source of anxiety for them, you must address that too.
If you try chess and your child becomes bored and restless anyway, don't give up a wonderful tool. Try chess lessons again from time to time. You may be very surprised.