White box replacements are almost like new. I have had excellent experiences with them. I understand you think your entitled to one by saying "Should", but if you don't have AppleCare or out of warranty, this is the likely your only option.
I've had great experiences with white box replacements. I think refurbs are basically like a brand new iPhone with extra quality control. I've had more trouble with brand new models out of box. Never had an issue with the white box replacements they gave me.