But he shouldn't. Black's g-pawn is worth more to White on the board than off it.

This advance really is not in White's interests - at least not unless and until the Black King has far removed himself from the vicinity. This he will have to do if Black is to effect a breakthrough anywhere. White, fighting merely for survival (i.e. a draw) has to try 32.b4, completely closing up the Q-side. The Black Bishop has no entry point except via the g7-b1 diagonal, after which it has no good targets (It would be an entirely different story if Black had his dark-squared Bishop instead!). Probably, if White were to play 32.b4 I would as black play 32... Bg6 33.a3 Be4 34.Kf2 Ke6 35.Kg3 but even then, if 35...c5, hoping to break up White's Q-side pawn phalanx, bxc5 creates a passed pawn, and Black still has to solve the problem of finding a way to break through. This seems achievable only by a second sac ...b4, which finally opens up the light squares through which the BK can penetrate. But by then the Black king strayed a long way over to the Q-side in search of the breakthough, Black can consider creating a second passer with h3-h4, and don't overlook the one at c5 that has been bypassed. The Bishop may well find itself trying to stop two wide apart passers. I'm inclined to conclude that the position after 32.b4, 33.a3 is pretty much drawn. Let's see what happens if Black too stubbornly goes for the win: 32.b4! Bg6 (...Bd7 is another idea) 33.a3 Ke6 34.Kf2 Kd6 35.Kg3 Kc6 36. Kf2 Kb6 37.Kg3 c5? 38.bxc5ch (38.dxc5ch is worth a look, too, since the BK is on the b- rather than the d-file) 38...Ka4 39.h4! and what is Black to do? 39...gxh4 40.Kxh4 and Black might actually be losing! The King can't extricate himself from the Q-side without he gives up another pawn by 40...b4 41.axb4ch, and even then White's g-pawn will cost Black his Bishop. I've gone into quite a bit of detail, but these kinds of considerations will crop up quite often in endgames.

Well, a passed pawn is inevitable, of course. I think what Thomas means is he can't allow a protected passed pawn in this position (as the h-pawn, once it advanced to h5 would be protected by the g-pawn). He's right of course, but the more critical corollary here is that, given the opportunity to open the position (to allow his King to penetrate the White position), Black must not let it go by. In that sense 3...gxh4 is indeed forced.

OK, but I prefer 33...Bg6 here, or even 33...a5. After 33...Bg6 White can give ground with his King, allowing Black to play ...Kg5, or move a Q-side pawn. If the latter, Black reciprocates, initiating a series of waiting moves until the Q-side is completely locked up, and then White has to give ground with his King: e.g. 33...Bg6! 34.a3 a5 35.b3 c6 36.b4 a4 37.Kg3 Kg5 38.Kf3 Bh7 (another waiting move) 39.Kg3 Be4 40.Kh3 Kf4 41.Kh4 Ke3 etc.

Even if White exchanges pawn for bishop -the most likely outcome here- who will be closer to influencing the real center of play, the queenside?

An endgame with more interesting features than you'd think!