Summary
You should not manually delete your content from iCloud while signed in with your Apple ID, as this will delete the content from the iCloud servers and any of your devices signed in to iCloud.
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Quick Start allows you to quickly set up a new device, however, you'll still need to restore from an iCloud backup to transfer the rest of your content and data.
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This will at least allow you to start using your new device right away until the backup to iCloud is completed on your old device.
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According to