Instead of tapping Charging, which takes you to the Battery screen as you've found, tap where it says USB Charging and you should have the option to change it to File Transfer.
We always want for you to have the option to transfer your phone content to your computer. I’m sorry that you’ve had trouble since the last software update on your phone. Let’s figure this out. Have you tried the suggestion given by tikibar1? Are you able to by chance have another computer to plug it into?
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