I don't think anyone answered the OP's question about the reason why they have such a dished face.
Don't take this as gospel, as this is what I'm remembering from my Equine Exercise Physiology lectures, but the dish in the face developed due to the hot, arid climate the Arab evolved in. The dish effects the shape of the sinuses, which, coupled with the large nostrils, allows a large oxygen intake and cools the air efficiently, helping prevent potentially fatal overheating of the brain.
Compare them to the draft horses/British native ponies - they often have a more roman nose. The larger sinuses allow the air to be warmed before it is taken into the lungs, as they are typically from colder climates.
Well, going back to my observations on Arab horses, and I prefer quarter horses, so I know only what I was told by my Bedouin about Arab horses, but I doubt that is the answer. One, Deserthorsewoman has pretty much given a good argument as to why that probably isn’t the case,. And as I said, none of the horses I saw through the desert when I lived there in Jordan had the dish face, a few of the fancy ones had a little, but mostly they were brought in from the north, Syria and even as far away as Turkey, all the well known local horses had pretty much straight faces, and they were the ones the Bedouin liked for their endurance races, they liked the thoroughbreds for shorter distance races. (on a side note I have a video I took of a race in which my friend’s thoroughbred won, it’s worth watching to see how fast this horse could go, I’ll try to figure out how to upload it and make a post about it).
The horses I saw there all look like Arabs, they have the same body and the face is kinda the same but for the dish. I’d almost be willing to bet money on it that the reason so many have the pronounced dish face is because some of them, in the past probably exhibited a predisposition for it, breeders focused on that as a particular trait of the breed and have selectively bread horses to emphasise that particular characteristic. The same has happened with all sorts of animals that are put in the show ring, they are bread for “breed characteristics” above everything else. For example, have a look at a real Shetland pony, then look at an American Shetland pony; hard to believe they came from the same stock.