Children's shows should be intellectually stimulating and promote the value of seeking to understand the world around us. That should be the main concern especially in contemporary American society. Our academics looks like a third world country's compared to many other developed nations.
Although there is a lot that can be meant by "mature" or "adult".
Older protagonists bear characteristics of a mature person in real life. For example, if I were to put a TV show with a 19 year old protagonist and his 22 year old wife and they both go on Debates.Org to think and write down their own opinions on topics. So, they would both intelligent and assertive. When the younger audience see this show, the characters would teach them to come up with their own ideas rather than being fed by the older people.
Younger protagonist with such characteristics would also teach the same thing to kids.
Children on shows today do seem "too mature", but also too obsessed with dramas that are really watered-down versions of adult problems. Every kid show now, especially on Disney, has three or four main characters, who are only worried about being viewed this or that way, dating, etc. We need more silly shows for kids. Shows that are funny and light. TV is a huge factor in how kids learn how to act now, and if all they see is kids worrying about things and concerned for their image, that's how they will learn.
If I understand your question, then this is something that many "children's shows" already do. Lots of live-action kids shows tend to try and mature the show's characters. For example, Boy Meets World started with kids that acted like normal kids, and kind of matured from there as the show went on, following the timeline of the same characters. Nowadays, a show about a preteen will over-exxagerate the maturity of characters to unrealistic levels, trying to make it seem like kids should be having mature discussions. I don't think that this is a universally helpful template for kid's shows, and I think that it contributes to a problem of disrupted childhood, for lack of better terms. Case in point: if I am a kid watching a show where kids my age are discussing adult issues, suddenly I feel immature, and want to rectify this by "leaping ahead" and disregarding my childhood in favor of more mature concepts. This can be unhealthy in some situations.