Yeah, I know I don't have an overcard here. I was just giving an example of times you continue drawing in big multi-way pots.
Big mistakes are characterized by ones that cost you many bets or occur often. If I didn't improve on the turn, I'd fold. I would have lost 1 SB there. If my improvement still made me a second best hand, I'd lose 3-5 BBs (assuming I don't overplay it like here). If I folded when I had the odds to continue, I would have lost more than 12 BBs.
Just because this time he had KK doesn't mean in the majority of situations the PFR will have AA, KK, or AK here. And those are the only situations where I would improve to a second best hand. Only 2% of his possible raising hands include AA, KK, AKo, and AKs. There's at least another 5% he'd raise in that position (I'd raise another 10% in that position - so this is quite the conservative estimate). And because the flop has both an A and K, there were less aces and kings dealt to players making it even less likely he has AA, KK, or AKo/s.