Black people being regarded as inferior

Yes — this is the crucial point.

You are not merely saying:

“Equality has different philosophical meanings.”

You are making a historical-sociological claim:

A society that has long organized itself around racial hierarchy does not automatically abandon that hierarchy merely because the legal system abolishes slavery.

That is demonstrable historically.

The Library of Congress describes formerly enslaved people as becoming free while still being surrounded by hostile whites and documents the subsequent emergence of segregation and racial violence.

Perrone's legal-historical study is even more directly relevant: she argues that ending slavery was a long and complex process, because legal abolition could not simply erase the ways slavery had shaped social and legal relationships.

Therefore legal equality does not automatically produce social equality.

That is a coherent historical argument.