I use chess.com. Their mobile app and website are both very intuitive, well designed and easy to use. Accounts are free and allow you to play as much as you want with friends/strangers. There are also a few bots you can play against, and you can do a few puzzles/day. The premium upgrade lets you play against more bots, do unlimited puzzles/day and has some handy other features, but it's definitely not necessary.
As far as I know, there is no way to block playing against strangers, and I would imagine your kiddo would do that if using the app unsupervised. Two main risks come to mind:
1. There is a chat feature where, very occasionally, people get nasty. You can change the settings to disable any incoming chats, but depending on how tech savvy your kid is, it would be easy to turn these back on. This would not be a risk if you're monitoring their usage and making sure they keep the chat off, but something to keep in mind.
2. When people are obviously losing, they sometimes stall to try to get you to quit, or otherwise try to trick you into losing games. If your kid is super competitive, this will definitely get on their nerves, but it may be a good teaching point to help teach losing gracefully.
Close supervision would allow you to avoid these issues and just play for free with the family you mentioned.
I've used the app for 2+ years, and I would be comfortable allowing my kid to use it if I could check in on them every 10-15 minutes. But, that decision is ultimately up to you. Let me know if you have any other questions.
