This is all horsefeathers. Every major chain has its off brand, store brand "value" line of staple products, and carries them alongside national brands. Consumers can choose lots of advertising for about the same thing or a lower price. Entirely up to them.
But, that's not the discussion here... LOL...
The discussion (per the article) is about Walmart discontinuing the competing name-brand, in place of their own brand. And that's what the other major chains and retail outlets don't do...
And it's precisely that -- which I think will be the one thing which will get a lot of their customers looking elsewhere (at another competing retail outlet) to get that discontinued item that Walmart no longer stocks, in place of their own "white" brand... :-)
That's not how I read the article. At most, the article--of questionable veracity--indicates that Wal-Mart will give more shelf space to the GV brand at the expense of name brands. Entirely removing name-brand products from their shelves would be tremendously stupid.
As an aside, though, there is definitely a niche to be had in selling private label products only. Trader Joe's, for instance, sells only private label products and its stuff is on par with, or better than, premium name brand products.