Why is the g3 and Bg2 plan so strong vs the Dutch? For example, in other main openings (KID, Gruenfeld, Queen's Gambit, Queen's Indian) it's just an alternative to the "classical" systems, those with no early Kingside fianchetto. A strong one maybe, but an alternative. In the Dutch, it seems that if White wants to get an advantage in the opening he can do it only with this set up. Nf3, g3 and Bg2.
Optimal piece setup in openings are determined by pawn structures. If you compare the common pawn structures of the openings KID, Gruenfeld, Queen's Gambit, Queen's Indian with the Dutch you will notice a substantial difference: The existence of the black f5-pawn and white's pawn on e2 rather than e4. The black pawn structure of Stonewall Dutch is usualy c6-d5-e6-f5 whereas the pawn structure of Leningrad Dutch is f5-e6-d6 (and c5 or c6 depending on circumstance). I will limit the answer to the Stonewall Dutch to be more precise however this should also be applicable to other openings of Dutch flavor.
Examining the stonewall formation following the principle of optimal piece activity - that is all pieces should be placed where they are most active - we can easily see that the white-squared bishop would not be placed actively on anywhere but g2. The only comparison could be with the d3-square (we can eliminate other squares quickly: c4 impossible, e2 too passive having little or no perspective).
Comparing the d3 and g2 squares the important difference is the bishop on d3 targets the kingside and g2 targets the queenside. We know for a fact that Black plans to play on the kingside which means her/his pieces will be placed on or near to the kingside. Following the principle of "attack your opponent's weaker spot/side" we see that bishop on d3 will not make much sense as it will, so to speak, "bite on granite".
Moreover, the bishop on g2 has multiple purposes: Putting pressure over the long diagonal and adding an extra defense for the king. In chess a piece is most useful when it is both attacking and defending, so g2 is not only the optimal square for the white-squared bishop on Dutch openings. It is the only square that actually makes sense.
The fianchetto of the light squared bishop is a "double barreled pistol" against the Dutch.
It is strong defensively against Black's king side attack. It is strong offensively in the center.
Most games resolve themselves into a "race." Here, the issue is can White get a winning position in the center before Black's king side attack breaks through?
The fianchetto is particularly good in the Dutch defense race. In the other games, Black is stronger in the center and weaker on the king side, so the advantage is less pronounced.