He did not know, he was just a mad and selfish idiot who had a tantrum. "I want my power back! Waaagh!" He assumed the secret was in the mom's body, and the kids hadn't pulled off any Titan transformations up to that point, so they needed an extra push.
I'm trying to figure out if there is any way to justify a case like Kuklo from the spinoff happening in your GoT timeline.
https://attackontitan.fandom.com/wiki/Kuklo . No special powers, but a good reason to turn the war against the undead into an eternal siege against the northlands. I was thinking pieces of his Titan body could shrink until eventually a little mindless Titan pops up… but nothing fit perfectly, unless you want Jon to tear his Titan body into pieces from the "savage madness" we saw influence Eren at the start of the series. Then the bad half could be a source of wild titans and traumatized humans. Or he could make a glow or Shazam-like--lightning spawn whenever he touches a living or undead Giant native to the northlands. It stuns him and somehow makes the situation worse, such as by splintering his power into a wild Titan or something.
What will wind up mattering are the Rules Jon, or his predecessor from when the wall went up, sets for the powers. If you go deep into GoT lore, you might be able to find hints of the Rules that governed Titan powers during the past age passed down into rules about the Seven deities or the ways society was arranged before dragon riders came over from Old Valyria.
