Same. I also checked the data usage on my phone and google authenticator has used NO mobile data since I've had it installed for over a year. So I'm calling bullshit on this article.
I also do not see nearly as many permissions requested as in that screenshot. It needs photo and video permissions because you can upload qr codes and stuff. Also you can (don't have to) link it your google account, so obviously it would have access to your google stuff.
I'm so sorry that you're dealing with this situation. Unfortunately, no, I would not go. It's not fair to you. I commend your dedication to your son but it would do too much damage to you than is worth it.