Yes, the problem is known to be cooked. The problem is actually a total win for White without a joke promotion.
Therefore, the task of creating a position in which the only way to win is to promote to a Black queen is an open one.
The problem can be found in the Die Schwalbe Chess Problem Database, or Schwalbe PDB for short. The solution engine there actually supports promotion to enemy units. The German notation means this: K=King, D=Queen, T=Rook, L=Bishop, S=Knight, and B=pawn.
The problem is that the normal promotion move 1. exf8=Q+, supposedly refuted by the study, actually wins for White!
[Title "Gyula Breyer, Magyar Sakkvilag, 1918, White To Play And Win"]
[FEN "1N1bknbr/2p1Pn1p/2K1Rp1p/4pP1B/p3p3/p3P1N1/P6P/8 w - - 0 1"]
1. exf8=Q! Kxf8 2. Kd7! {This is the move that Breyer missed!}
The problem is for Breyer’s study is that after 1. exf8=Q+ Kxf8, the move 2. Kd7 more or less leaves Black in a zugwang.
Furthermore, after the joke promotion to a Black queen, the position is indeed a draw as you noted. Stockfish analysis confirms this.
[FEN "1N1bkqbr/2p2n1p/2K1Rp1p/4pP1B/p3p3/p3P1N1/P6P/8 b - - 0 1"]
1... Qe7 2. Nd7 Qxe6+ 3. fxe6 Be7 4. Nf5 Kd8 5. exf7 Bxf7 6. Bxf7 h5 7. Bc4
The inherent problem with the promotion to a black queen is that Black does not have to play, after 1... Qe7 2. Nd7 Qxe6+ 3. fxe6, 3... Ke7? as Breyer claimed in his analysis. This ends up in a loss for Black in the form of a mate in 5 moves.
(Note that the move 1... Be7? loses to a mate in 5: 1... Be7? 2. Kxc7 Qg7 3. Nxe4 Qg1 4. Nxf6+ Kf8 5. Nbd7+ Kg7 6. Ne8#)
[FEN "3bk1br/2pN1n1p/2K1Pp1p/4p2B/p3p3/p3P1N1/P6P/8 b - - 0 3"]
3... Ke7? {This is an incorrect move, contrary to Breyer’s analysis} 4. Nf5+ Ke8 5. Kb7 Be7 6. Kc8 Bd8 7. e7 Bxe7 8. Ng7# {This is the shortest mate afterward as found by Stockfish}
The correct line for Black to play is 3... Be7! and White has no mating net, thereby drawing the game, as given in your line I showed at the top of this answer.
It is a mystery exactly how the ingenious Breyer missed this possibility in his analysis. The most likely answer, through analysis of Black’s options after 4. Nf5, is that Breyer thought that Black must give up protection of their c7 pawn through a bishop move.
Black has 6 legal moves after 4. Nf5-5 by the bishop and one by the king. The moves Bf8 and Bd8 both lead to a mate in 5: 4... Bf8 Kxc7 Bd6+ 7. Kc8 Bc7 8. Nxf6+ Kf8 9. e7# (among many possible lines) and 4... Bd8 5. h4 Be7 6. Kxc7 Bd6+ 7. Kc8 Bc7 8. Nxf6+ Kf8 9. h4#. Bb4 and Bc5 both lead to shorter mates in 4 with the same basic ideas.
The move 4... Bd6 looks like a tempting refutation but White still wins even after that move. Here is one such winning line for White.
[FEN "4k1br/2pNbn1p/2K1Pp1p/4pN1B/p3p3/p3P3/P6P/8 b - - 0 1"]
4... Bd6 5. h4 Kd8 {Anything else results in a mate in 4} 6. exf7 Bxf7 7. Bxf7 Kc8 8. Nxf6
The end result is a positional win for White, although any normal person would struggle win of course.
The last, only non-bishop move 4... Kd8, therefore, is the ultimate refutation in this study after the joke promotion that Breyer somehow miscalculated.
TL;DR: In conclusion, Gyula Breyer’s joke study is cooked on two faults. A normal promotion actually wins after the line 1. exf8=Q+ Kxf8 2. Kd7!, and Black is in zugzwang
Secondly, in the joke promotion lines after the moves 1. exf8=Black queen Qe7 2. Nd7 Qxe6+ 3. fxe6 Be7 4. Nf5, Black can play the only non-bishop move 4... Kd8! and draw the game.
Overall, the study is refuted by a king move from each side. Therefore, it seems like Gyula Breyer could not, to some degree, properly analyze king moves in the endgame!
