That video segment never mentions "AI", and it never says that the halting problem has been solved -- it explicitly says it's unsolvable. What it says is that solutions to the halting problem could be verified under a certain computational model.
That's obviously an extremely rough summary of what the paper shows, but I'm not sure how one could do better in a 3-minute summary for a non-expert audience. And based on Scott Aaronson's response to the paper, it seems like calling it a "landmark" result is very reasonable:
> Still, assuming this one stands (as I’m guessing it will), I regard it as easily one of the biggest complexity-theoretic (and indeed computability-theoretic!) surprises so far in this century. Huge congratulations to the authors on what looks to be a historic achievement.