What does the "Transfer Using Quick Start" do?
Despite the description on Apple's website, Quick Start is not a distinct method for transferring content from an old to a new iOS device. It's simply a way of skipping a couple of the more laborious screens in the setup assistant on the new device. It is still necessary to perform the usual steps to ensure that the backup from which you choose to restore is fully up to date. (In particular, the instructions to first unpair any Watch because this "automatically backs up" the Watch, are incomplete at best.)

Using Quick Start, you will skip the following setup assistant screens on the new device:


Choosing a Wi-Fi network
Entering your Wi-Fi network password
Entering your Apple ID for iCloud
Entering your Apple ID for iTunes and App Stores, if you use a different one


Using Quick Start, you will need to complete the following additional screen on the new device:


Entering your old device passcode (which you may have to do anyway depending on your setup, regardless of whether you use Quick Start or not)


Quick Start has no effect on how data is restored. You must still have first created an up to date backup, and select that as the source from which to restore.