Whether he can cbet the flop for value heavily depends on you guys 3-betting AK pre or not. It's definitely not terrible to check it. On the turn he's not betting for value or trying to deny someone the odds to draw. - he's blockbetting to make his own mega draw and doesn't want to get raised off it.
His real mistake here imo is not realizing that once the hand goes to the turn, his pair of kings becomes a decent value hand and he blocks or dominates all draws. He should just go ahead and pot it.
River is just obviously terrible.
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