Positional Question in the English Opening

On move 12, Stockfish wants to play Nxb6. I see the point in that. Afterwards it places the pawns on e4 and d3, shutting in white's light squared bishop...I don't really understand that. The LSB had a wonderful diagonal, so why should e4 be played?
I can think of preventing black from playing d5, playing against the only remaining Bishop from black (by putting every pawn on white). Are there any other reasons? The weakened d4 square is not a problem for now because neither knight can make use of this hole.

Is this the only setup white should strive for in this kind of position or are there any other plans to consider?

Thank You!

Not an expert in the English, but at first glance:

The open diagonal sure looks nice, but do you really NEED your LSB to control it right now? Your pawns will do that just fine. And you won't do much to b7 or a8 at the time being anyways. You might eventually trade your e-pawn for Black's f-pawn, too, if Black tries to free himself, or push d4 at a later point, so the "loss" of the diagonal is not necessarily permanent.

On the other hand, d3 and e4 make your DSB happy. You could go for Bb2 + d4 too, but this will carry some risk for the then backward c4 pawn (in contrast to the actual game, the c-file is half open!), an eventual f4 break looks safer.

It's just trading an advantage for another. By playing e4 and f4 while clamping the center through d5 White gets prospects of a kingside initiative - it's the routine plan for Black in the King's Indian and other openings like some Sicilians and Retis.

But in this position any sensible setup will do - this is just a bad position for Black. I am quite the rookie in the 1. c4 e5 English but it seems to me you should only play Nc6 in reaction to Nf3 by White. Nc6 Nd5 then Bb4-c5 when White still has the option to play e3 Ne2 is already slightly better for White and worse, quite hard to play for Black (stats are not good in basically all lines, even when Black let the bishop on b4 to speed up development).

After the usual 1. c4 e5 2. Nc3 Nf6 3. g3 Bb4 4. Bg2 0-0 instead of Nc6 the main line sees the Bg2 filing another complain for bad treatment : 5. e4!?, with the same setup as in the Rossolimo Sicilian, see Carlsen's games.

First , I want to say that I would never do what the engine does. It seems to me very unreasonable to block Bg2 and surrender Black squares. I assume all this has to do with Black's inability to find good counterplay after all this happens.

lichess.org/study/exYNNVfF/PnJAXag2

This is the line engine gives and it gives a huge advantage to white, something like +1.9 I can't say I understand why white is so much better but trying to understand the position I realised that Black has no plans. Except putting a knight on c5 there is nothing else he can do. Q-side is blocked , the centre is blocked and the only available break is f5 but that creates a huge dilemma. If Black plays it he allows Bg2 to become a very strong piece.If he doesn't play it , he has nothing else except moving his pieces around aimlessly until white does something. It's not a surprise that Black actually tries to exchange the "bad" Bg2 with Qc8-Bh3 in some lines. White has all the time in the world to prepare an f4. Perhaps the point in engine's "thinking"(there is no "thinking" of course , only infinity calculation) is that White willingly blocks Bg2 to prevent Black from creating counterplay with the one and only way he could, f5. White denies Black's any counterplay as compensation for his bad piece. It doesn't sound bad. The more I look in the position the more I like it. Of course , needless to say , I would never think of doing that.