did i miss value on river ? this is a satty

[Guido]

UTG: 9,565 (64 bb)
MP (Hero): 9,750 (65 bb)
MP+1: 13,420 (89 bb)
CO: 9,490 (63 bb)
BU: 9,815 (65 bb)
SB: 6,029 (40 bb)
BB: 6,680 (45 bb)

Pre-Flop: (330) Hero is MP with K♣ 9♣
1 fold, Hero raises to 352, MP+1 calls 352, 4 players fold

Flop: (1,034) T♣ Q♥ T♥ (2 players)
Hero bets 689, MP+1 calls 689

Turn: (2,412) J♠ (2 players)
Hero bets 1,608, MP+1 calls 1,608

River: (5,628) 8♣ (2 players)
Hero checks, MP+1 checks

Total pot: 5,628

Showdown:
MP (Hero) shows K♣ 9♣ (a straight, Nine to King)
(Equity - Pre-Flop: 42%, Flop: 25%, Turn: 75%, River: 100%)

MP+1 mucks A♥ 4♥ (a pair of Tens)
(Equity - Pre-Flop: 58%, Flop: 75%, Turn: 25%, River: 0%)

MP (Hero) wins 5,628

[Guido]

solver says

preflop i should raise a little bigger

flop as played with my betsize betting has a little more EV than checking but I should betting 25% potsize 74% of the time and checking 26% of the time

I guess this is because here I have no real range advantage and no real nut advantage

but i have a big enough stack and enough equity to call a potential normal raise sizing of villian

as played on turn when hitting the middle straight solver says betting and checking has almost the same EV

(in game I was unsure whether i should check or bet and again which sizing)

solver says 50% potsize is the optimal size here

but solver also says my betsize has same EV

(solver only knows 25% potsize , 50% potsize , 100% potsize and 200% potsize)

on river solver says if I want to bet I want to bet pretty much all betsizes are OK but 50% potsize would be the default

I should then check 85% of the time

and bet 50% potsize 15% of the time