That part can absolutely be inferred. When in both cases the child ended up with a random woman on some backwater planet, you have to assume that’s because those are the most common type of women in the galaxy. Especially when in both cases Palpatine easily could have given the child to a woman he trusts.
I said this at some point, but I really like the idea of the Emperor needing them to fight each other over the throne because they’ve both had visions of themselves on it. Obviously neither of them really wants that, but Rey refuses to cooperate with Kylo’s alternative plan. So when they fight on the Death Star wreckage, we play a slight motif from Duel of the Fates. They are literally dueling each other until one of them dies, fulfilling one of those visions. It’s only because of Leia’s intervention that neither of them dies and they both are in the light still.
I’ve got a few other slight changes I’d like to add to your version, hinventon. But I think the ones you did make are solid, so I’ve kept most of them:
“I pushed you in the desert because I needed to see it… I needed you to see it… who you are. The dark side is your birthright, Rey…”
“You’re lying.”
“I’d never lie to you. Your parents were no one. Junk traders expecting a child they never wanted.”
“Don’t!”
“But no father of yours had any part in your conception.” (people most associate Anakin with having no father)
“I don’t want this!”
“…just like Vader.”
“No!”
“You’ve seen it, too, haven’t you? Destiny.”
(Rey sees the vision of herself on the Sith throne)
“Your parents saw you as unnatural… they feared you.”
“Stop talking.”
“The same way Luke feared me.”
(Cutaway to heroes capture)
“The Force showed me glimpses of a girl like you long before we first met. When we fought, in the forest, the power inside of you awakened. You didn’t reveal yourself to me alone… you were exposed to the Emperor… just as he’d intended.”
(Visions of Rey beating Kylo on Starkiller, falling into the dark cave, etc.)
“No!”
(A pedestal shatters and the mask of Vader falls to the ground)
“So that’s where you are.”
“Before you chose the Jedi, Palpatine wanted you alive."
“No…”
"I’ll come tell you why.”
“Rey… I know the rest of our story.”
“Tell me.”
"The Emperor influenced the Force itself into creating life. First my grandfather, and then you. You were his chosen heir. You… are a rightful Palpatine.”
"The Emperor only wants us to fight each other, so his most worthy heir can claim the throne. But he hasn’t realized the strength of our connection, Rey… a bond, not of blood, but of raw power in the Force. We can end him and the Sith, and create a new order, together…”
(The most important changes I’ve made here is to make the references to the prequel trilogy as clear as possible so that as many people as possible understand what’s happening).