CF, there really isn't such a thing as generic stuff.
It's literally another product in different packaging. You might not like the other product as much, but it's not some garbage thing. Publix doesn't have an ice cream factory. It's literally another brand and they pay less to slap their label on it.
While you are right Publix doesn't have an ice cream factory in *some* case the generic/store brand has lower standards. Sometimes this is a big deal sometimes it is not.
I knew someone who worked in a bakery the national brand as did one of the store brands required that all the chocolate chip cookies be perfectly round and unbroken. Two of the store brands allowed for up to 10% of the cookies to be broken or deformed. One of the store brand only required that 50% of the cookies be round an unbroken. All the cookies tasted the same and was made at the same time but the generics got the "defective cookies" In my opinion that isn't a big deal. But on the other hand generic batteries that are from batches that didn't meet the standards of the name brands is not something I want. Costco and Target have high standards for their generics. Walmart is a mixed bag. Dollar tree is mostly stuff that doesn't meet the standards of everyone else.