Quality binning is a thing, but not quite like that.
If they do a factory run for "Brand A", adding the label is a component of that run. There's three main outcomes:
- The product passes Quality Assurance / Quality Control, and sold to mainstream retailers
- The product fails QA/QC, but for cosmetic reasons only such as marring on the packaging or label misalignment. These get sold to discount retailers such as Big Lots[1], but still under the main brand.
- The product fails QA/QC due to a reason that makes it ineligible for public sale. Depending on the margins of the product, it may or may not be worth some level of re-work to make it eligible for sale within one of the two bins above. Otherwise, it's either donated (if the product is safe but ineligible for sale due to some regulatory reason such as consumer protection laws around the word "New" on labels) or scrapped (and might be sellable for some industrial use) or destroyed.
But in all cases, it's still "Brand A" and not sold as a completely different inferior brand.
Private label runs are different. A retailer will come to you with a specific set of requirements around ingredients, price points, profit margins, etc and you'll design a formula that matches their requirements[2]. While the manufacturer is leveraging the same machinery, product formulation expertise, and procurement capabilities, the end result may be completely different than any product you actually sell in your name brand portfolio due to differences in priorities and market goals. For example, Costco, Walmart, and Dollar General may be relying on the same manufacturer for their private label trashbags. But decisions on the thickness of the plastic for the bag, whether to use "break protection" patterns on bag design, whether to incorporate draw strings or just little flaps of plastic, whether to package it in a cardboard box or plastic sleeve, etc are all made in coordination with the retailer requisitioning the private label and will end up as a distinctly different product than both the private label runs for other brands as well as the manufacturers own name brand formulations.
Ice cream is a really big one here. You'll notice that it's always sold by volume (fluid ounces) rather than weight (ounces). Manufacturers control the volume of air that gets incorporated into the ice cream, and can use the same amount of product by weight to produce a pint, quart, or half gallon of ice cream. Both within the name brand portfolio of the company and the private label runs, you'll have wildly different end products based not only on the ingredients chosen for a specific formulation but the density chosen as well.
[1] These discounters also tend to get inventory that is perfectly fine, but that didn't turn fast enough and you're willing to let go at a discount to get it out of your warehouses.
[2] You'll also see this within some name brand portfolios as well. You'll see this in the form of retailer-specific model numbers for something. This happens for lots of reasons, from Amazon having specific Alexa functionality incorporated into the version of a product to Walmart demanding a compromise on warranty period or part quality so that they can get to the retail price point they determine is most optimal.