What are the benefits of pushing the h pawn?

> What are the benefits of pushing the h pawn?

Attack on the kingside.

At least, defending g5 is an advantage. Later on, it will be difficult to stop without weakening your pawns.

And then, I hope you weren't thinking about 50..Kxf6. If so, you would see the real benefit.

as LMLance5500 says, attack on the K-side.
Look at how that pawn, once wedged into h6, discombobulated black's K-side fianchetto. But if black had prevented the advance with 7... gxh5, the position on the K-side would be opened for white's attack. Something that was telegraphed by Be3 and Qd2, and that black has done little to prevent or prepare for.

I don't know (or care) what Stockfish recommends, but 6.h4 earns an exclamation mark from me.

For comparison, look at Borkowski-Nunn, Groningen 1974. 1.e4 g6 2.d4 Bg7 3.Nc3 d6 4.f4 Nf6 5.Nf3 0-0 6.Bd3 Na6 7.e5 Nd7 8.h4
In Borkowski-Nunn, black is able to counter attack white's centre, and the game is entirely double-edged. (It's a really good game to review, just for the tactical fireworks) But in this game, white is entirely free to do as he pleases.

Gets players out of book;
Burns a tempo, to not lead the game;
Gives the opponent a head start to see their plan;
Offers Black the center first;
Creates hope to castle Queen side;
Makes the game more challenging;
Breaks chess principles on purpose to make chess entertaining.

Everyone has a reason for a move. The true reason for a move can only be confirmed by the player that did the move.
The rest are just assumptions until proven other wise. Engines only give their mathematical point of view. Like I gave my biased point of view. All my resourcefulness and instinctive moves come from experience.

So it doesn't feel left out

So you don't have to think of a better move which probably wasn't a better move but a blunder.

this is awesome I actually learnt smn

Hey, #1:
Step 1: Become a 2200
Step 2: Do whatever you feel like

@verylate said in #4:
> as LMLance5500 says, attack on the K-side.
> Look at how that pawn, once wedged into h6, discombobulated black's K-side fianchetto. But if black had prevented the advance with 7... gxh5, the position on the K-side would be opened for white's attack. Something that was telegraphed by Be3 and Qd2, and that black has done little to prevent or prepare for.
>
> I don't know (or care) what Stockfish recommends, but 6.h4 earns an exclamation mark from me.
>
> For comparison, look at Borkowski-Nunn, Groningen 1974. 1.e4 g6 2.d4 Bg7 3.Nc3 d6 4.f4 Nf6 5.Nf3 0-0 6.Bd3 Na6 7.e5 Nd7 8.h4
> In Borkowski-Nunn, black is able to counter attack white's centre, and the game is entirely double-edged. (It's a really good game to review, just for the tactical fireworks) But in this game, white is entirely free to do as he pleases.

What if black obstructs the pawn with ...h5