"Josephine's mother told her she was the "daughter of the Prophet Joseph Smith, she having been sealed to the prophet at the time that her husband was out of fellowship with the Church."
What did she really mean by this statement? That she was Joseph's daughter by right of her mother being "sealed" to the prophet? I am sealed to my mother and my step-father... and have taken him as my "father" vs. my biological father. I was not produced by a physical relationship between my mother and step-father... but became his "daughter" through marriage and moreso, and by my mother and I being sealed to him in our eternal family...
I see how you could make that interpretation based on that part of the quote, but we should look at the rest of her statement.
This leaves her statement with only one candidate for what this secret was - that she was the daughter of Joseph Smith in the ordinary sense of the word. If Joseph was the biological father of her child, Sylvia Sessions's words are exactly how we would expect a 64-year-old woman in 1882 to say it. I'm having a hard time imagining what additional explicit detail she might have felt would be necessary, considering she was unaware that Mormons living 126 years in the future would want to read other meanings into her simple statement.
If this affidavit were evidence that undermined the doctrine of the church, I could understand the desire to spin it, but as Will pointed out, it doesn't matter whether Joseph's polyandrous sealings were physically consummated marriages if God commanded him to do it.