Your objective assessment of White's position after 6. Ng1 is quite on the mark.
A. Objectively speaking...
It's hard to see how this position could be good for White.
On the plus side -
- White has a good center.
- Black's pawn on f4 is weak.
- Black's kingside is a bit loose.
On the minus side -
- White is a pawn down at the moment.
- White's own kingside is weak.
Is a good center worth a pawn? Sometimes it is, sometimes it is not. In this position, it can be shown that at best White is only equal. There is no line that gives White an advantage.
B. Subjectively speaking...
White played the King's Gambit because White wanted a sharp tactical game.
On the plus side -
- The White player has successfully achieved a complex tactical position which suits his/her style of play.
- Complex tactical positions may not suit the Black player's style of play. Perhaps he/she played 1...e5 with the intention of getting a quieter position.
- If the White player is better at tactical positions than the Black player, White has greater chances of winning the game.
- The White player may be better prepared in the opening than Black player for this particular line.
One example - Nigel Short, a master of tactical positions, outplayed his strong Grandmaster opponent Vladimir Akopian in this position, playing White.
On the minus side -
- Complex tactical positions may suit the Black player's style of play and Black might enjoy these complications.
- Black might be as good as White in tactical positions or even better. Thus, White has good chances of losing the game.
- Black might be very good at defense and at best White will only manage a draw.
- The Black player might be prepared just as well in the opening as the White player or even better in this particular line.
One example- the great champion of the King's Gambit and a master of tactics, Boris Spassky himself, played this position both with White and with Black, winning the game with Black.
It seems to me that White's reasons for going into this line are more subjective than objective. The computer evaluations all favor Black slightly, but that's expected because Black is materially a pawn ahead and computers often are biased towards material. Objectively speaking, here are some lines which show that White doesn't get anything special out of this opening. In my opinion, in the line after 6...f5!?, White is even slightly worse.
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]
[Event "6...f5!? and Black might be slightly better"]
1. e4 e5 2. f4 exf4 3. Nf3 d6 4. d4 g5 5. h4 g4 6. Ng1 f5 (6... Bh6 7. Nc3 c6 (
7... Nf6 8. Nge2 Nh5 9. g3 O-O 10. Bxf4 Nxf4 11. Nxf4=) 8. Nge2 Qf6 9. g3 f3
10. Nf4 Qe7) 7. Nc3 fxe4 8. Bxf4 (8. Nxe4 Qe7 9. Qe2 Nc6 10. c3 Bf5 11. Nf2 Bh6=+
) 8... d5 9. Qd2 c6 10. Nge2 Nf6 11. O-O-O Be6 12. Kb1 (12. Ng3 Bd6) 12...
Nbd7 13. Ng3 Bb4 14. a3 Ba5 15. Be2 Bc7 16. Rdf1 Bxf4 17. Rxf4 Qe7 18. Rhf1
O-O-O=+
One sample Grandmaster game in this line -
Hector vs Peter Leko 0-1 1995
4. Bc4!? is a perfectly playable alternative to 4. d4. The point is that Black has to "waste" a move playing 4...h6 first and cannot play 4...g5 directly. If 4...g5, then White get's a better position after 5. h4! as follows -
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]
[Event "Black cannot play 4...g5 after 4. Bc4!?"]
1. e4 e5 2. f4 exf4 3. Nf3 d6 4. Bc4 g5? 5. h4! g4 6. Ng5 Nh6 7. d4 Bg7 8. Bxf4+=
However, I should mention that after 4...h6 and 5...g5, Black is very solid in the mainline resulting position and it's not easy for White to break through.
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]
[Event "4. Bc4 h6!? 5.d4 g5 and Black is doing well"]
1. e4 e5 2. f4 exf4 3. Nf3 d6 4. Bc4 h6 5. d4 g5 6. O-O Bg7 7. c3 Nc6
Are there reasonable non-mainstream deviations from this line?
Considering the above discussion, I think a reasonable non-mainstream deviation would be to play 4. Bc4 and 5.h4!?, preventing g5. Objectively speaking, White is still slightly worse because of the material deficit, but subjectively speaking, the position is quite complicated with plenty to play for. One idea for White's would be to hold on to the strong pawn center, castle queen side and attempt to put pressure on Black's f4 pawn with moves like Ne2.
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]
[Event "5. h4!? is worth a try"]
1. e4 e5 2. f4 exf4 3. Nf3 d6 4. Bc4 h6 5. h4 Nf6 6. Nc3 Nbd7 (6... Bg4 7. d4
Nh5 8. Ne5! dxe5 9. Qxg4 Nf6 10. Qf5 Nc6 11. dxe5 Nd4 12. Qxf4 Nxc2+ 13. Ke2
Nxa1 14. exf6 Qxf6 15. Nd5 Qxf4 16. Bxf4 O-O-O 17. Rxa1+/-) (6... c6 7. d4 b5
8. Bd3 Nh5 9. Ne2) 7. d4 Nh5 8. Qd3
6.Ng5!? h6 7.Nxf7, with Kasparov winning, chessgames.com/perl/chessgame?gid=1070825. It is more aggressive for sure.