So you raise the river and what hands you beat are gonna call you? AJ maybe. All weaker hands will likely fold. All better hands call. You talk about playing optimally. Checking behind is the Optimum play.

Betting the river is just pointless. Your max value arguement has no ground because your not getting value from a bet because weaker hands arent calling. Lets say you put 4K into the pot and hes put 4K into the pot. Your hand has showdown value so you check behind. If you have a better hand you win 4K, If he has the better hand you lose 4K. Lets say this time you bet 2K on the river to put him all in. He folds his weaker hands so your 2K is returned and you win 4K . If he has a better hand he calls and you lose 6K. By your own admission there is like 1 hand possibly 2 in which he calls with a weaker hand and you win 6K.

So he has 1 possible hand that makes a river bet profitable. Thats assuming hes even calling with AJ.