I'm pretty sure my phone is affected, but according to that page:

    To prepare your iPhone 6s for the battery replacement process, please follow the steps below:

    Back up your data to iTunes or iCloud.
    Turn off Find my iPhone
    Erase data and settings in Settings > General > Reset > Erase all Content and Settings

Just to replace a battery??

They swap out the entire phone, and send the old one out to be refurbed and sold.

Not anymore, at least the NYC stores don't. They take your phone in and give it back to you in a couple of hours(depending on how busy they are). That's how they did it when I had mine replaced last week. The 5th Ave store didn't have any batteries in stock so I had to either wait a few days or go to a different store that had them.

You should never send a phone with your personal data out in the wild. If an Apple Genius is going to take your phone to the back of the Apple store and out of your sight, back it up and restore it to factory settings.

They might be sending you a refurbished phone instead of replacing the one in your existing phone

It's always prudent to backup your data if your device leaves your possession. What if FedEx loses the package or it's stolen off the truck?