Tony explained why d3 is a good move. Brian explained why "hope chess" is bad in general, so I'll answer the question, "Is 1 ... Kg7 itself a bad move, and why?"
In chess, the your most valuable commodity is moves. Every move must work toward your victory. If you play a chancy move and it doesn't work, your opponent is strengthening his or her position. At a critical time, this can turn a crushing victory into a draw, or a draw into a crushing loss.
The short of it is, Kg7 is as almost bad as giving away a bishop for nothing.
I chose the move 1 ... f5 from your position. Like Kg7, it forgoes ... d3 and allows white to blockade the pawn. Unlike Kg7, it achieves 4 separate goals of strategic value. Afterward, I let Next Chess Move (running stockfish on 1 CPU core) play both sides. After f5, black is an anaconda suffocating its prey. White has no truly good moves for the rest of the game, and only a handful of self-directed moves.
[Event "?"]
[Site "?"]
[Date "?"]
[Round "?"]
[White "?"]
[Black "?"]
[Result "?"]
[FEN "r1bqr1k1/pp5p/2nb1p2/2p5/Q1Pp2pN/1P4B1/P3BPPP/4RRK1 b - - 0 1"]
1... f5 {consolidates the pawns, expands the king-side pawn space, and forces:}
2. Bxb6 Qxb6 {the trade favors black due to the strength of white's bishop. In addition it comes with tempo. Essentially, black gets a free move of development.}
3. Bd3 Bd7 {In this line, Bd3 comes too late. Black plays Bd7 and unites the rooks, forcing white to concede the open file. Further, the x-ray attack on the queen will produce another move with tempo.}
4. Rd1 Nb4 {Knights are strongest on the 5th and 6th ranks with pawn support. The knight now controls several squares inside of white territory.}
5. Qa3 f4 {f4 was a goal of f5. Pawns are strongest side-by side, and this pawn-space strangles any thoughts of white activity on the king-side.}
6. Bb1 Re2
7. Qc1 Nxa2 {White's pieces are stuck on defense. The only piece with immediate offensive prospects is the white bishop.}
8. Bxa2 Rxa2
9. g3 Re8 {Black activates the remaining rook. All of black's pieces are in dynamic positions and black's pawn space is suffocating. From here, things just get worse and worse for white.}
10. Ng2 f3
11. Nf4 Bf5 {One of few positive moves for white in the game.}
12. b4 Rc2
13. Qa3 c5xb4
14. Qxa7 b6
15. Nh5 Re7
16. Qa4 Rce2
17. Qa8 Rd7
At this point, Next Chess move's evaluation is hovering around -5.0, meaning that black is functionally a rook ahead. It doesn't stop there. White has no prospects, while black has two passed pawns and active, cooperating pieces and plenty of space to work in. Black ultimately queens two of the pawns, one to win material and one permanently.
In comparison, here's how Kg7 plays out:
[Event "?"]
[Site "?"]
[Date "?"]
[Round "?"]
[White "?"]
[Black "?"]
[Result "?"]
[FEN "r1bqr1k1/pp5p/2nb1p2/2p5/Q1Pp2pN/1P4B1/P3BPPP/4RRK1 b - - 0 1"]
1... Kg7 {evaluation goes from -3.6 to -0.77, a full piece in difference.}
2. Bd3 Bxg3 {The trade is forced, or black will lose the bishop and the game after 3 RxR.}
3. h2xg3 Bd7 {This time, black is playing without the tempo gain from f5.}
4. Qa3 Rxe1 {The trade momentarily concedes the open file to white.}
5. Rxe1 Qf8
6. Qc1 Re8
7. Bf5 Qf7 {Black's pieces are stuck on defense. Black is successfully contesting the open file, but the position is nearly equal.}
8. Qd1 h5
9. Rxe8 Bxe8 {Now black has no more active pieces. The positions are still nearly equal.}
10. Qc1 Bd7
11. Bb1 Ne5 {One of the few positive moves for black in this game.}
12. Qa3 Qf8
13. Qxa7 Bc6
14. Qa5 d3 {Black finally threatens to advance the pawn, but it's much too late.}
15. Qc7+ Qf7
16. Nf5+ Kg6
17. Nh4+ Kg6
Draw by repetition. White can't let up, or ... d2 wins the game. Black can't move anywhere but Kg6 and Kg7, or the queen is lost.
So that is the harm of moving 1... Kg7. It gives white time to stabilize, equalize, and find a draw in a game where black should achieve a crushing victory.
You should notice that many of the moves are found in both games. But the slight change in timing changes everything about the outcome.
