But these aren't unauthorized follow-up payments, a person you are responsible for and have logged in with your credit card credentials is sitting and making actual purchases.
I doubt a court would see it this way. I'm sure it wasn't the intent of the account owner to authorize his kid to make any purchases.
Suppose a 10 year old walks up to a cashier at a Walmart, dumps 50 candy bars on the belt, and hands the cashier a credit card with no adult in sight. The cashier rings it up and charges the card. The kid opens all the candy and gives it away to friends, eats it, whatever. Later the adult discovers that the kid took his card out of his wallet when he wasn't looking and complains to his credit card company.
The fact that the kid had the card in no way authorizes its use. In fact, a court would laugh at a cashier not questioning the use of a card by a 10 year old.
In that case, the store could decide to file a criminal complaint against the child. Merchant agreements aside, the clerk has no way of knowing if the parent authorized the use of the card, or even if the card isn't the child's' although a 10 year old would be a bit of a stretch. They accepted it in good faith, if the parent claims fraudulent use then the store could attempt to recover from the child.
I realize your 10 year old example is a bit extreme but it's still fraud. However, plenty of parents let their kids use their cards,and stores accept them without question. Most parents wouldn't claim fraudulent use been if the kid took it without their knowledge; so is it the store's fault they assume the person using the card is the an authorized user? I would like stores to check my ID with any purchase since that would make it a lot harder for someone to use my card but most stores don't want the hassle and many customers would get upset as well.
This is really an extension of that. The phone owner did not give his kid permission to buy something, and thus it was not an authorized use of his account. The fact that the phone can't detect this scenario in no way makes the owner responsible. You can't unconsciously give somebody permission to do something. A computer might misinterpret your actions as authorizing something. A company might write a bunch of contract terms that claim that you can authorize something non-explicitly. However, in the end the only thing that matters is actual intent. If you don't intend to authorize a transaction, then no contract exists, and thus no obligation to pay the bill exists.
Except Google has no way of knowing if it is the child or the adult using the phone, so it's reasonable to hold the owner responsible in such cases.