Pawn and Two: <outplayer> Burn had a strong advantage until 30.Qa4??. In the tournament book, Steinitz referred to 30.Qa4?? as an extraordinary blunder. As the time control was 15 moves per hour, time pressure may have played a factor in Burn's error.
Additional analysis is needed to determine White's winning chances at move 30, but he had a number of very promising moves available, including 30.Rc1, 30.Qb3 and 30.Re1.
You have indicated that Burn could have held the draw with 57.Kb4. Interestingly, Steinitz also reviewed 57.Kb4, and he concluded that Black would win, beginning with the move 57...Ka6.
After reviewing this, I have to agree with Steinitz. After 57.Kb4 Ka6, my computer indicates the best line is 58.Kc3 Rd1 59.Bh4 Ka5 60.Bf2 Kxa4 61.Kc2 Rf1 62.Bg3 Kb5 63.Kd2 Kc4, and Black will win. Also, in other continuations, White is unable to prevent Black from capturing the a-pawn, and then maneuvering his pieces for further gains.
Even if White imprisons the Rook by 64.c6 Ra1 65.c7 Ra8 66.Bh4 Rc8 67.Bd8 Kxd4, the ending is a win for Black.