It'll be something really obscure and difficult to diagnose such as trusted installer failing to correctly set permissions on a particular registry key. 

HTH.

Removed him from the family, removed all accounts except a local admin account. Installed Chrome through that, then added his account back in. Checked Chrome was working on his account (it was) then added him back to the family. Seems to have fixed it, for now at least.

Glad I was able to help.

The web is full of people with Microsoft Family Accounts problems and seems to have been for several years. I hadn't even heard of them until your post.

I wasn't aware of them in the context of windows, but it's handy for sharing stuff on the Xbox. Allows him to have his own account, save games, friends list etc but not have to buy all the downloaded content separately (or have a separate Xbox live subscription etc).

It also let's me set how long and when he can play, who has is great when I'm feeling mean :'D