Are they at the "Lovenstein Institute"?
Andrew Jackson's parents were born in Ireland (Scots-Irish). That makes it very unlikely that he had African-American ancestry.
The story about Harding has been around for a long time. It was spread by a professor who hated him. We still don't know if it has any credibility at all, but it may be one reason for Harding's insecurity. It's been said that many early settlers of the Middle West may have been part-Black and blended into the White population over the years, but who can say whether they included Harding's own ancestors?
The story about Coolidge is based on the lack of knowledge of his maternal grandfather's ancestry and the man's last name: Moor. That's not much. But for what it's worth, here's Coolidge's mother, Victoria Josephine Moor Coolidge:
Again, who can really say?
I think Harding himself had the last word. He didn't know himself - how could we know?