My son has wrestled in a singlet that my wife purchased while I was gone last year. I think they missed a couple of practices prior to the first tournament when the club singlets were given out. I was very pleased to see what she bought him, a singlet with the exact same colors as the team singlets, just a little different design. He is proud to be part of the club he wrestles for, just like he is proud of being a member of his Jr High wrestling team.
Wrestling is definitely an individual sport when it comes time to strap on the ankle band. As he learned from his first year in Jr High, it is a little more special when you are part of a team where your individual points really do contribute to the overall success of your team. I recognize that team points are kept at Kids' State as well as some of the other Kids' tournaments. It really isn't quite the same as it will be when they are competing for their schools.
I have no problem with kids wearing singlets that are not from the club they wrestle for. When the time comes, and they are wrestling as part of a team where their individual results contribute to the outcome of the teams' success, I believe they will be very proud of wearing the school colors.