I have some issues with this. Let's first contextualise Ariana's death. Snape cast the curse that killed Dumbledore, but Snape's soul wasn't torn apart because murder wasn't Snape's intention. If Dumbledore did cast the curse that killed his sister (can't be confirmed), her death would have been completely accidental even if there was a murderous intent, as the intent would be for Grindelwald. This intent can't be confirmed anyway because we don't know which curse was even cast, it might not have killed Grindelwald but it may have killed Ariana because she was young, defenseless, and weak. But honestly, none of this matters anyway because the way to repair one's soul is remorse, which is clearly what Dumbledore was feeling!
Why would the piece of Dumbledore's soul be transferred to Harry from the wand when Voldemort tried to kill him in the forest? This doesn't follow the horcrux logic in the rest of the books. I'm so far unconvinced that Dumbledore would even consider making a horcrux. It's serious dark magic that ruins your life. You have to relish the fact that you've murdered someone and your afterlife is spent in pieces and in pain (look at Voldemort in Limbo!). Perhaps it's because Voldemort made so many, but you would have to assume that there would be consequences after just one--which incidentally we do not see with Dumbledore in Limbo.
I like the bit about fragmented souls and the fluidity of memory between them, but we also know that memory and knowledge is fluid between magical images. Dumbledore's portrait continues to inform Snape and the Order, and he keeps track of Harry after his physical body dies. I feel that this is proof enough that spiritual Dumbledore could keep track of Harry without making a horcrux. And like I said above, I'm still super unconvinced that Dumbledore would choose to do such a thing.
But again, none of this matters because Dumbledore spends the rest of his life in remorse, meaning that a horcrux would be impossible to make.