I believe this is a great competition among children. It encourages creativity and artistic displays which are all too often lacking in public schools today. It encourages children to work hard and never give up. I think the only downside could be parents that get overly involved or push their children too hard, but even with that I believe it does a lot more good than harm.
The competition seems to be one that is positive. For example, the 2014 winner did a drawing that depicts water purification. The prize for her was a $30,000 scholarship. While I'm not always a big fan of competitions like this since you can never know if kids received help or if they completely came up with a concept and design on their own, and art is very subjective, if Google wants to do this as a means of giving back, then in that way it's definitely positive.
Google Doodle should continue to host this competition. It supports alternative opportunities for young children to be creative, as well as give incentive for them to do so. Google Doodle is an activity that takes children away for electronic devices and encourages them to use their own imagination. Additionally, it is alternative way for them to be competitive outside of sports.
Yes, Google should continue to host a doodle competition among children, because it is good for the kids to learn to compete. Children have the excitement of working hard and competing. Kids need to learn how to win and lose gracefully. They also need to learn that hard work pays off in life.