This app was mentioned in 3 comments, with an average of 2.67 upvotes
Hope this isn't against the rules. I'm the dev of an app called Toddle that might be what you're looking for. It plays Youtube videos and locks the screen. You can choose what videos get played via youtube playlists so there's no chance of something inappropriate popping up.
There's a new one called Toddle that has pre-selected videos.