So I need to buy a ticket from nice to the border and then a ticket from there to Genoa?
No, I said https://www.trainline.eu/ will sell you tickets from Nice to Genoa. Some trains are direct (no change), some involve a change, but in either case Trainline will sell you a ticket end-to-end.
What I said was, IF you choose one of the options that involves a change of trains, then neither of the national sites would sell an end-to-end ticket, because neither sells foreign tickets, that is why I recommended Trainline.
And it is way too early to buy tickets. Tickets after the change in December will only appear late October or in November.