“Smith is not saying this happened to formerly enslaved people” vs. “Smith uses the United States as his example”

There is no contradiction once we distinguish the level of the claim.

Smith does two things:

A. His general analytical proposition

He distinguishes legal citizenship from lived citizenship.

That is a conceptual distinction that can be applied beyond one particular historical episode.

B. His empirical case

He uses the United States as the historical/political case through which he develops or illustrates that framework.

“He is not saying only: ‘This happened to formerly enslaved people after the Civil War.’” he is not limiting his conceptual framework to that particular episode.

It does not mean:

“Smith says nothing about the United States.”

He does discuss the United States.

Therefore, you can legitimately cite Smith for a proposition about the United States when the proposition is actually supported by the part of Smith you are citing.

The important distinction is:

“Smith discusses the United States” but it does not mean that “Smith makes every historical claim about the United States that I want to make.”

For example:

Supported by Smith's framework:

Smith distinguishes legal citizenship from lived citizenship in the United States.

But if you write:

Smith argues that the Thirteenth Amendment failed to produce social equality among formerly enslaved people because white Southerners retained racial attitudes inherited from slavery.

you need evidence that Smith actually makes that specific argument, not merely that he discusses the United States.