Theory connecting Amaranthus to Jamie's Voice Over in the finale
Or more precisely, his last will and testament. I spent several hours gardening this weekend without a hat on so it's entirely possible that I baked a few brain cells but also, I've been thinking about this since I watched the episode.
Wills are one of the most helpful historical documents we have because they are typically contain a boatload of information about relationships, wealth, and power. (In American history, wills are one of the most reliable ways the descendants of enslaved people can trace back their ancestry because their ancestors were often passed to enslavers' children via wills.) There was no real expectation of privacy to speak of - you put the information in the will because you want your survivors to know and that likely means information will be publicized.
I had the same reaction as u/hello-8282 here - Jamie's just putting William's business on main?!? But then I had to chop up a dead lavender bush.
I think it's because of Amaranthus. She seemed to come out of nowhere and go nowhere. But, my probably-sun-damaged theory is that she was a catalyst for William publicly accepting Jamie as his father. Because of her presence, he was repeatedly asked to (re)consider the boundaries of family and what it means to claim family members. Combined with the conversation with Brianna, and the poor guy was deep in the family feels.
I think the writers had Jamie provide William's full name because he had permission to do so. I don't think we necessarily needed to see the conversation where that happened because if you listen to how Sam read it, there's a lift in Jamie's voice. Which, to my ear, was meant to communicate pride. In effect, "he's my son and I can say it freely."
Bonus theory: I think the way they handled Richardson entire plot line was a way for the writer's to try and remedy some of the badness in Season 4 with regards to slavery. (But also, I haven't read the books.)