The fact that he’d suddenly be having real-time Force skype sessions with her would be unsettling regardless of if he was having brief visions of her in the past. And, as we know, these sessions don’t stop with Snoke’s death, who claims to have instigated them, so clearly there is more happening here.

Also, he can still be having visions of a girl like her only to find out she’s nobody of true importance except for what she can potentially offer him - thus he believes that’s why she was being shown to him. He wouldn’t mention it in TLJ because it goes against his argument. It’s only in TROS when Palpatine tells him about the fact that both Anakin and Rey that he puts two and two together to realize that they share the same power in the Force, so it makes sense that the Force would be showing her to him. He shares it in this conversation at last because it bolsters his own argument that they are destined to work together.

Ultimately, you aren’t giving me any alternatives here for something he could say in this part (you didn’t give enough content in your last draft for the space here).

I’m willing to take a look at the bit about her parents only because I think giving any sort of characterization to them is a bad idea. But I fundamentally disagree with your part about Vader. It’s not in the “lore” it’s directly stated by Shimi: “There was no father.” We need to invoke this decades-long mystery again so that we can answer it, and this is the only way casual fans could possibly remember it. The way you currently have Kylo describing things sounds like adoption.