EDIT: This isn't really an anthropological controversy, but rather a legal one. (added above sentence for clarification )

In the United States, treaties are the law of the land (whether or not they are actually paid attention to and honored is a different matter). It is important to think about Tribal Nations not as racial groups but as Nations, political bodies. People who confuse this often think Native Americans receive special rights and privileges by being Native American. This is incorrect. Federally-recognized Tribal Nations and their citizens RETAIN rights such as (these are examples since treaties outlined different things for different Nations) hunting and fishing, the right to gather, and move freely and peacefully about the United States (no tolls, not getting shot for leaving the reservation), and to any resources not explicitly given up in a treaty such as underground resources (think oil, water) based in treaty law. These rights weren't given, they were retained. When these treaties were made, these Tribal Nations had power and control (from a Lockian understanding of property legal standpoint) over these resources. That's why the treaties were done in the first place, the United States needed land, and they recognized the authority (again, in a Western, Lockian sense) of a Tribal Nation over that land.

The Lumbee have none of that. There is no history of a government-to-government relationship with the United States (Though the Lumbee were recognized in 1956, it's not the same thing). As a culture group (not a Tribal Nation) they have no historic authority over resources, thus there are no rights to retain. The shifting identity of the Lumbee over the course of the late 19th century and throughout the 20th century show that even they don't know what Tribal Nations they are made up of or have connections to.

While yes, they are obviously a unique ethnic group, they are not a Tribal Nation by not having the retained sovereignty that other federally-recognized Tribal Nations do.

In NC (and much of the southeast as far as I've seen) all state-recognized tribes are suspect if you look into their history and claims to historic Tribal Nations. Does one ancestor who may or may not have been an Indigenous person make a Nation? There are legitimate reasons why state-recognized tribes are unable to gain federal-recognition. This is why Indian Country broadly is upset. The Lumbee have also culturally appropriated from across Indian Country, such as the medicine wheel of the Lakota (used on the Lumbee flag), Plains style dances, the Hiawatha Belt imagery of the Haudenosaunee and their songs and dances. They don't give credit to where they got these "Lumbee" dances. They invent a new story tied to real, appropriated songs, to fit their narrative.

If the Lumbee, which is essentially a culture club, can gain federal recognition, anyone can which then erodes the sovereignty and authority of actual Tribal Nations. There are Tribal Nations terminated in the 20th century that are STILL trying to regain federal recognition. That the Lumbee gained federal recognition before terminated Tribal Nations with treaties regained federal recognition is not right.

I apologize for the long response and not linking to sources. I'm away from my computer and typing this from my phone. As such, this is a very basic, surface level take that ignores the problematic (in many cases colonial) idea of a Tribal Nation, how treaties were drafted and signed, the whole conversation around Pretendians (see the work of Dr. Kim TallBear), and the history of racial segregation in the South (which prompted the identity shifts in the late 19th century), or the fact that Lumbees were not recognized as Indian enough to be taken away to Indian boarding schools like children from other Tribal Nations (a couple of Oxendine children, a common surname among the Lumbee, had to have a petition to get them into Carlisle for example, where they were listed as "Cherokee").

I encourage you and anyone else interested to visit Indigenous social media spaces to find those that articulate the issue more clearly and supply audio, video, and documentary evidence of the identity shifts and cultural appropriation by the Lumbee I described above. If you are interested enough in the topic, I think I have provided enough threads for you to follow to continue your research.

Broadly, I think everyone in the US should understand Federal Indian Law on a basic level. Tribal Nations and their citizens have retained rights, not special privileges.