I'm unsure.
A significant quantity of stuff sold with the Kirkland label is just whitelabeled, with varying degrees of hidden-ness. Their recent beer for instance is labeled Kirkland up top, but also have the company (Hopf Malz something), and it's a pretty trivial amount of digging to get to the underlying Gordon Biersch.
I don't know where you draw the line though
* white labeled but not really because it's still got the original name on it
* white labeled without the original name
* self produced
Probably you could get finer grained than that too.
So it's entirely possible that you could pass a law banning large chains from doing whitelabeling, but actually you'd still get the same stuff on the shelves with their original labels and it'd be fine.
Or it's possible you could pass a law banning chains from selling their own produced stuff, but still have whitelabeled "house brand" be ok. But then if you were equitable about that in online space, - amazon couldn't sell fire phones or echo dot alexa things, but they could sell an "amazon choice" if it's just whitelabeled. So idk.