The vision Rey has in the cave in TLJ could also be seen as the Force telling Rey she has no real parents, she sees only herself.
This stuff writes itself lol.
But yeah Kylo would definitely want to emphasize the potential dark side of her destiny, the trouble would be doing that without having him lie in such a critical moment. Even Vader doesn’t lie to Luke. I tried a new line in my previous comment, but anybody else can take a crack at it, too.