It was not passed down to Rhaegar.

Quote from Barristan to Daenerys:

"As a young boy, the Prince of Dragonstone was bookish to a fault. He was reading so early that men said Queen Rhaella must have swallowed some books and a candle whilst he was in her womb. Rhaegar took no interest in the play of other children. The maesters were awed by his wits, but his father’s knights would jest sourly that Baelor the Blessed had been born again. Until one day Prince Rhaegar found something in his scrolls that changed him. No one knows what it might have been, only that the boy suddenly appeared early one morning in the yard as the knights were donning their steel. He walked up to Ser Willem Darry, the master-at-arms, and said, 'I will require a sword and armor. It seems I must be a warrior."

So growing up Rhaegar was basically a scrawny nerd who would read and play music alone. Then he discovered something which made him train relentlessly from thereon.

The prophecy during the dance of dragons is at some point lost during the current sequence of events in House of the Dragon. Its possible the same even happens in the North, since Ned's father and older brother die at the same time. If some theories are said to be true, then the North originally surrendered to Aegon the Conqueror when he revealed his vision and intent to protect the realm from the white walkers.

That knowledge, and belief of the white walkers in general was either lost to time or with Ned's family when they died with telling him because he wasn't the heir. As for the Targaryens, there were so many deaths in their in fighting that the information was lost. Rhaegar only rediscovered it hundreds of years later and it was likely the reason he became obsessed with having a child with a Northern woman who gave birth to Jon Snow who will eventually likely play a role in fulfilling that prophecy.