Yes perhaps... But this response is based on knowing what villain held. This response is biased because you know villain had q-9. It is not necessarily the best play always.
In real time, there is a range only. Hero is not going to know what villain holds, for sure. It is never going to be bad to bet the river here with K-K. Donks will often raise with busted draws and 9x hands. If they fold, big deal.
Holding kings full I would hate to have villain check behind.
No one except the most extreme aggrofishtards are going to raise the river on a missed draw at these stakes, in which case you will never fold an overpair regardless of the board.
You can claim I am biased in my assessment, but I wasn't referring to the Hero there, it was about the villains choice to call. Clearly the villain agrees with my assessment of the situation. So I know that he thinks along the same lines in this scenario, I feel enough so that I can justify the proposition that he would bet a nine here.
Assuming we know nothing we can still conclude that a check raise is better than a donk (at least I think so) from my analysis below:
As for the Hero's perspective, I never donk the river so my default move would be to check/raise anyway. However I cannot imagine a villain who wouldn't bet a nine here, clearly he doesn't check behind near-nut hands, we know this because he was willing to bet the turn. From that we can gather he is not exceptionally passive. So we gather he bets a nine, but we cannot know if he is foolish enough to raise a donk with trips. So we gain the same value from check raising as we do donking whenever Villain has a 9 (-15% max missed value taking into account bet sizing in the worst case scenario).
The problem now is how to extract value when he doesn't have a nine, I can't think of the villain calling a reasonably sized donk bet (like the one hero made) with anything less than a nine, so checking gives him the opportunity to fire everything that isn't willing to call. That is the only way to extract value from other hands here, if villain is particularly fishy he might call a half-pot donk if he had TT or something but again he is probably going to make a half-pot sized bet with TT anyway. So you miss no value by checking.
