As pointed out by Michael West, 1...Nf2 provides White the opportunity to exploit White's overwhelming advantage whereas 1...Ng3 is a forced checkmate in three (i.e., Black has a guaranteed win).
Black is down too much material and has no compensation if they cannot force mate. After 1...Nf2??, Black is utterly lost.
Stockfish 14+ NNUE confirms this (+9.8, depth = 27, 8115k nodes/s). Despite the low depth, the example lines provided illustrate the point.
[FEN "8/2Q2pk1/3p2p1/NP1Np3/4n2p/7P/6PK/r7 b - - 0 1"]
1...Ng3! 2. Qxf7+ Kxf7 3. b6 Rh1# {Black wins} (1...Nf2?? 2. g4 hxg3+ 3. Kxg3 Ne4+ 4. Kf3 Ng5+ 5. Ke2 Ne6 6. Qxd6 Kh7 7. Nc4 e4 8. Qe5 Ra4 9. Nf6+ Kh6(2. g3 Ne4 3. gxh4 Ra2+ 4. Kg1 Ra1+ 5. Kg2 Ra2+ 6. Kf3 Nd2+ 7. Ke3 Nf1+ 8. Ke4 Ng3+ 9. Kd3 e4+))
Note: Diagram from Black's point of view to match OP's diagram.