Regarding games schedule, if the user is not root you can use cron or systemd timers, but this could be more user friendly for a child:
Create a zenity, xdialog, or any kind of graphical dialog explaining why it's not gaming time (even remembering the allowed gaming time) and put it in /usr/bin/games-denied
Rename the games to /usr/bin/GAME-allowed
Then with cron/timers just symlink /usr/bin/GAME to one or the other, so when the kid click in the game will get a dialog or the game, depending on the hour.
Also, I could recommend:
1) Try to automate the re-installation (disk images, preseed/kickstart for the OS, ansible for the setup...)
2) Provide some way to make backups (the kid will create content, you may want to save)
3) Download a lot of content (drawings to colorize, songs, funny images, whatever may interest this kid) and put it in the Documents/ dir so the kid can explore. Install extra fonts, icons, and whatever may improve document creation.
4) You may explore the system package manager for educational, science or artistic stimulation packages. Kids are surprised when they see the can play the piano in the laptop, or they can see the stars, or ... :)
apt-cache show ubuntu-edu-preschool
apt-cache show ubuntu-edu-primary
apt-cache show ubuntu-edu-secondary
5) Put a sound recorder icon, and a cam recorder icon, right in the desktop (if there is sound card or web cam in the laptop), you may get surprised by their creativity :)