Pawns pushes in the center There are some situations in chess where a pawn push in the center is the most played move, and also the one suggested by engines. However this is not done for winning material, but just for trading pawns. One such example arises from the Philidor Defense:

rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1
1. e4 e5 2. Nf3 d6 3. d4 exd4 4. Nxd4 Nf6 5. Nc3


3. d4 is the move I'm talking about, and the game after that move usually continues like in the PGN above.
I don't fully understand the advantages that White gains by exchanging its d-pawn for Black's e-pawn, ending in the position shown above. They are both central pawns controlling central squares. Can someone tell me some insight or considerations I'm probably missing?
The main ideas behind d4 serve several purposes.


It threatens dxe5 winning material.
It opens up your pieces, like the Bc1, in particular.
It gains space since the trade on d4 leaves white with a pawn that is more central than black's d6 pawn.
And probably most important, and this is typical of many
double-king-pawn openings, and that is that it is easier to inhibit
black from playing the freeing d5 move. If he manages that move
soon, he has generally solved all his opening problems, and the two
center files will open up, and the major (heavy) pieces will be
traded.