Why wasn't this strategy chosen? In the below position, White chose to activate his Knight with Ne4. Of course, he will have to play g4 to prevent …f5, and that requires the moves Kh1 and Rg1. These were the moves played in the game.

r2qrbk1/1pp3pp/2n1bp2/p2np3/8/PP1PPN2/1BQNBPPP/R4RK1 w - - 0 1


In the game, Black played Qd7-f7 and Rad8. Although this does put pressure on the b3 and d3 pawns, it cannot be sufficient to warrant a capture. This also weakens Black's hold on the a5 pawn.

Instead, Black can play Nb6-d7-c5, attacking b3 and -- more importantly -- d3. This also places the knight on a better square, as it was doing nothing on d5. An alternative is also Nb6-d7 and f5-f4. What recourse does White have here?
The game you refer to is Bobby-Fischer - Ulf Andersson, Siegen 1970.

It is famous for Fischer introducing the impressive plan Kh1,Rg1,g4,Rg3,Rag1 that has often been used since in hedgehog position, mainly for Black.


  White chose to activate his Knight with Ne4. Of course, he will have to play g4 to prevent …f5


Actually ...f5 is pretty hard for Black to engineer in that kind of position: he has to prevent Neg5 and Nc5, to provide support for his Pe5, and he won't be threatening ...e4 nor ...f4 any time soon. Note that the Re8-Bf8 dispositif is better suited for queenside play while keeping a solid f6-e5 center, the plans would be different in a position with Bb6 and Rf8 for instance.

Fischer's g2-g4 is actually is much deeper plan than just securing the Ne4: it more generally grab space on the kingside (e4 but also f5 for a Knight, g3 for a rook) and threaten g4-g5 at a suitable moment to open the long diagonal for the Bb2 and the g-file for the rooks.


  Black can play Nb6-d7-c5, attacking b3 and -- more importantly -- d3. This also places the knight on a better square, as it was doing nothing on d5.


This is reasonnable enough, but White can hold on b3 and d3 without too much trouble. In later years, Qb1-Bd1-Bc2 would even become fashionable, even if it probably wasn't in Fischer's mind back in 1970. This regrouping secures both pawns and adds pressure on the diagonals toward the kingside when the game opens up, most often with d3-d4.

Moreover, after White's Ne4, it is not sure that Black Knight will reach c5 at all. Supposing it does, it would still be sensitive to tempo-gaining freeing move like b3-b4 and d3-d4.

Finally, I beg to differ about the Nd5. Although it is screening the pressure on the d-file, it is not "doing nothing": control of b4, e3 and f4 stops White from trying the most agressive plans with b3-b4 or f2-f4. Generally speaking, central squares that your opponent's pawns can't control are considered good spots for your knights.


  An alternative is also Nb6-d7 and f5-f4.


This however is a very bad plan. Even if Black was granted four free moves to achieve it without White moving at all, it would result in passive pieces (Nc6,Nd7,Re8) holding to the backward pawn on e5 that will be firmly blocked (Ne4) and under fire (Bb2, Nf3, possibly Qa1 or Nc4). 

And what do you try to gain with f5-f4 ? An attack on Pe3 ? For that purpose you wouldn't want to retreat your Nd5 to d7. Opening the f-file or the a7-g1 diagonal ?  Well, you have Re8-Bf8, not Rf8-Bb6. I see no reason for White to be afraid of f5-f4.