I’ve heard that sometimes the store brand is made at the same factory as the brand name brand, just packaged differently.
I don’t want to support Kellogg’s at all now, not even by buying differently packaged store brands.
Typically the big brands do NOT want to cannibalize their own brands by running store-brands as well.
Store brands are usually produced by other 3rd party companies who focus on low cost, no advertising, working with supermarkets, etc...