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There is absolutely no way I would let Google (or any other browser provider for that matter) have a copy of my Credit Card Details.

Nobody is forcing you to do that. Your browser is very likely going to have a "Don't remember" option.

You're free to keep your credit card details in your password manager or type it out every time.

You do realise that storing them (be it Google, LastPass, or any other hopefully trusty manager) is safer than carrying them around on you, right?

The parent is concerned about privacy, not security.