I actually love the white box replacements. I've found that they go through a really good QA.I’m not sure their interpretation of what they consider ‘new’. Replacement iPhones can be considered ‘like new’ that arrive in a white box without any accessories, but it’s possible it may have been previously used. Apples replacement devices have been thoroughly inspected, which usually includes a new battery, display and outer casing. What I would do when you do receive a replacement device, assuming that’s what they choose to do, is inspect it for any issues.
I've had two white box "Like new" devices and they were immaculate.
Less small issues than the originals.