This is a distinction without a difference. The mere fact that they don't own the factory producing the goods doesn't change the fact that they are acting as both producer and distributor in the same marketplace.

This is a huge distinction. They don't simply not own the factory, they don't produce the goods under any legal definition of the term. Nor are they a distributor under the legal definition of the term; they are a retailer which has specific and different legal meaning. (A distributor sells products or other parties that sell to end-customers. The manufacturer is usually the distributor of its own products.)

If Costco buys bottled water from Nestle they are only a distributor, and there is no conflict of interest. When they put Kirkland Signature water on the shelf next to it, there is a pretty clear conflict of interest that allows Costco to leverage it's power as a distributor into pricing power against producers.

No, it doesn't. Costco pays Nestle or one of Nestle's competitors for Kirkland Signature water. Absolutely none of the Kirkland Signature products are made by Costco itself other than the assembly of food in the food court.

Additionally, there is only a single buyer for the product line, regardless of whether the producer is legally independent. This is a vertically aligned operation - if Costco stopped selling Kirkland Signature tomorrow, it would cease to exist.

Note: by vertically aligned you actually mean "vertically integrated" (and I should have edited my earlier comment to correct the terminology). A vertically integrated company must own at least two of the following levels: the suppliers, distributors, or retailers of its products. (If it is a retail company, that would mean it needs to own one of the other levels of companies to be vertically integrated.)

The fact that they are buying the store brand products from the same producers who make the name brand product makes the entire operation look more aligned and concentrated, not less.

No, it's exactly backwards. The manufacturers don't just sell to Costco. They sell to other retailers as well. Vita Coco, for example, sells white-label coconut water to both Costco and Kroger (i.e., Ralphs), among other retailers. (Vita Cocoa used to be a client of mine.)

The entire dynamic as it exists gives chain stores negotiating leverage against those name-brand products, allowing them to get better prices on goods than a smaller, independent operator would be able to get.

Yes, that is how market power works. A participant with greater market power can use that as leverage to negotiate pricing. This is an acceptable use of market power; for example, it is the basis behind insurance pools, coops, group discounts, etc.

It would not be acceptable if the big retail chain used its market power to effect how a supplier deals with other parties or in other markets, such as by attempting to restrict a supplier from selling to competitors. (But note: exclusives are fine, so long as the exclusive product is not a condition of, or conditioned upon, restrictions on the sale of non-exclusive products, because a manufacturer can simply refuse exclusivity and offer it to competitors exclusively or non-exclusively or can negotiate the terms of the exclusive product with the retailer before them. See, for example, Orgain's nut-based protein powder, or the myriad models of TVs that are "exclusive" to each retailer.)