In reference to this blog and an entire Facebook Black Archaeology group that believes this.
I question this based on DNA, Bering Strait Theory, Native Americans do not all look the same and well.... proof. I've heard (Black) people who believe this say:
"Our identity was stolen and was given to someone else."
"If you had family belief of part Cherokee blood, that only means you are 100% Black (because Blacks are Natives)".
"The don't teach this because they want us to just be slaves."
Zheng He was brought up but there's no record of him landing in the Americans to my understanding. And Asians didn't come to the Americas until the 1700's. Giovanni da Verrazzano's description of the Native people is brought up also. And especially the Olmec heads as they appear resemble typical African features.
As you point out, DNA studies rule this idea out completely. So many of these theories are based in remarkably regressive concepts. They fit nicely in a long colonialist narrative that robs Native Americans of their agency and individuality. Assuming race based on appearance is incredibly outdated. Never mind that the descriptions are 500-year old first impressions from people who have a completely different conception and construction of race than us.
I've written extensively about one particular theory of ancient Africans in America here.
And Asians didn't come to the Americas until the 1700's
Hasekura Tsunenaga and his entourage traveled to Mexico in 1613 on his way to Rome, and again in 1619/20 on his way back to Japan. That's the first official expedition to the Americas by an Asian nation (excluding the regular trans-Bering voyages by the Yupik and other Arctic people) that I know of. Of particular relevance to this discussion is that this Japanese delegation was described in Rome as being "a color which borders on black." The terminology of the day isn't the same as what we use now.
Giovanni da Verrazzano's description of the Native people is brought up also.
This is likely due to a translation error, of a sort. While we no longer have Verrazzano's original letter describing his voyage, it's presumed to have been written in French, which wasn't his first language. Two accounts derived from the original letter, describe the people Verrazzano encountered as either "black like Ethiopians" or "brown like Saracens." The latter is more reasonable, and the going theory is that Verrazzano actually described the people of North America being like the "Maures" or Moors, which could cover a wide variety of skin colors, being about as specific as the modern phrase "people of color." When his account was published in Italy, Carli interpreted this as "black" while Ramusio went with "brown."
Verrazzano the Explorer is an old book, but it opens with a discussion of this topic. I can't think of a more modern work that takes the issue seriously enough to even bother debunking it. It also mentions the bit about the Japanese delegation being described as black.