That’s true. Palpatine is the source of this information, and Kylo is incentivized to believe it without question. As for the scenes on Ahch-To, those are about Rey’s lineage anyhow, and her being claimed as a Palpatine by name. There isn’t such a place to shoot down this news as it pertains to the Skywalkers. Leia would only fear the darkness that she associates with Vader in the flashback anyway, so it wouldn’t be a direct confirmation.
That being said though, have you seen the sly look Palpatine gives Anakin as he says “influence the midichlorians to create life?” It’s pretty wild. And what the opera show they’re watching looks like is a literal egg being surrounded by sperm cells.