Understanding when it's OK / not OK to trade off the fianchettoed bishop

Question

In this position:

[FEN ""]
1.e4 g6 2.d4 Bg7 3.Nc3 c6 4.Bc4 d6 5.Nf3 Nf6 6.O-O b5 7.Bb3 b4 8.e5 bxc3 9.exf6 Bxf6 10.bxc3 O-O 11.Bh6

According to the lichess database, the most common move here for Black is Re8. However, the engine says Bg7 is the best move, and both games in the Masters database in this line continue with Bg7.

Can someone explain the logic to me of why Black would want to trade off his fianchettoed defender here? Isn't that something you normally want to avoid, since it would weaken the dark squares around his king?

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On a related note, I've also come across this position while playing the King's Indian Defense:

[FEN ""]
1. d4 g6 2. c4 Bg7 3. Nc3 Nf6 4. e4 d6 5. Nf3 O-O 6. Be2 e5 7. Be3 Ng4 8. Bg5

The first time I saw this position, my instinct was to play Bf6 since my bishop is pretty horrible compared to White's, being blocked in by all of my central pawns. However, this just loses a pawn after 9.Bxf6 Nxf6 10.dxe5. My question is, would it be desirable for Black to trade off the bishops if doing so didn't lose a pawn? e.g. In this position:

[FEN ""]
1. d4 g6 2. c4 Bg7 3. Nc3 Nf6 4. e4 d6 5. Nf3 O-O 6. Be2 Nbd7 7. O-O e5 8. Be3 Ng4 9. Bg5

Why is f6 still more common than Bf6?

Answer 1

In the first position you mentioned, I'm guessing the reason why the engine recommends Bg7 is because Re8 weakens the f7 pawn, which could conceivably get loose later on.

In the last position, before your last line I was going to say that I liked f6 there rather than Bf6. I'm not completely sure why, but I think the reason is that the bishop is awkward on f6. It's blocking your f-pawn, and also stranding your knight on g4, which is now trapped. I don't immediately see how white can take advantage of it, but doesn't seem harmonious.

It seems like you're trying to look for general rules. Chess is a very concrete game and what is best depends on the specific position. There are some good general principles, but not a whole lot.

Answer 2

A weakness is only a weakness if it can be tortured by the opponent.

In example 1, White has already castled, so Re1-e4-h4, Q-d2-h6 seems to be the only way to bring on some scary firepower against the king. Also, the Pe7 still protects f6, so at least not a double weakness (Ph2-h4-h5-h6, Qf6 would be otherwise the standard plan). Still, the computer yawningly played e6 (to make the white bishop completely look on granite), h6, Rh8, already with a -1 advantage for black.

Concerning the KID black bishop, the KID is only becoming really KID-ish as soon as White plays d5. Good luck forcing White to d5 without this bishop! (It would be interesting to set up a position with the classical f4 attack of Black, remove the two dark-colored bishops and compare engine judgment, but I'm too lazy...)

To mock a famous saying, bad bishops have good later-on uses.