Hero calls the river.

Hero is so far ahead of Villain's pre-flop range after the flop. Only KK, 99 or 33 are ahead of Hero's hand at this point (almost definitely not KK with no PFR or 3-bet). Broadway cards have a pair and/or gut shot draws on this board, plus all Ax and Kx diamond flush draws. All of which are in Villain's range. (CPFR stat would be a little useful to narrow pre-flop range.) With all of this, Villain's C-bet means nothing with 5.75 AF. Raise on the flop is obv. (I'd have raised to $0.90, personally)

Turn is a brick and Villain re-C-bets. Usually, Villain has a piece of the board, but nothing they're confident about. A raise here is likely to take down the pot. As you noted, a raise to $1.75 - $2.00 would have been my choice.

Villain calls turn bet? Villain either thinks we're bluffing a draw and that his Kx, 9x, A3, or 43s hand is good, or that we have Kx, 9x, or 3x and his draw is big enough to take us down when it hits. (Or the lucky SOB has a set and is playing Hero like a fiddle.)

Unfortunate 3 on the river. Villain knows that not much in Hero's pre-flop range has a 3 in it. Villain might feel pot committed and that his only way to win is to bluff. Villain shoves.
Hero has top 2 pair and great showdown value. With so much in Villain's range that misses, and the weak bet on the turn, Hero calls.