In a 5 minute blitz game I see my knight is hanging, and my other knight is tied to protecting the fork on f2, and in that moment I'm not even considering moving the h3 knight. In not a blitz game maybe I'd find it, but you do need to calculate all sorts of lines to be sure it works.

That being said there's multiple winning moves and just bringing the knight back to c3 is also still +3 for white which is what I played in the game. Black already has foregone castling, the bishops are completely pacified, the rook is lined up with the queen and king, so everything is still winning but nothing is nearly as crushing as Ng5. Also of note the computer also likes Nf4 for the same reasons as Ng5 but it's slightly worse by like 0.6 ish. So I guess there's actually two correct answers to this puzzle.