Native dogs were not a breed then so they cannot be a breed now. To restore a breed there has to of course be a breed to begin with, but also there has to be remaining dogs to breed off of. He is trying to recreate a breed that never existed as Native American dogs weren't exactly selectively bred, therefore were not purebred.
I am well aware of La Flamme and heard about his "breeding program" growing up, that is my home town and he is a family acquaintance. I'm not here to tarnish his name, from what I know he's a decent man and his intentions are good. As I got deeper into the dog world it became more apparent that Indian dogs were an aesthetic mix to create what some may think a Native American dog may have looked like. Even with that goal in mind I don't see being achieved, his dogs are mixes of many breeds therefore lack consistency. As far as I know his dogs also are purely companion dogs, not working or sporting. He uses working breeds which means plenty of these dogs may not turn out with suitable companion temperaments. He does not utilize any modern health screening options out there which is key to ethical breeding. I don't think he even keeps hand written pedigrees, so keeping track of the lineage is non existent and likely contributes to the inconsistency of his dogs.
If someone is trying to create a breed there are many steps. Consistency, breed purpose and a clear goal come to mind.
Does it bother me? Not really, it hasn't gotten much traction and seems to be fading out. Even his website talks about breeding backward so Im not sure what you expected the DNA results to show. It's important to do thorough research when purchasing a dog.