Desert Eagles, do they serve a purpose? Absolutely. Do they serve every purpose? Absolutely not.
The point of the design was to bring magnum level power (using existing magnum ammo) to a semi auto handgun.
Yes, they are useful hunting guns. And for certain types of sporting competition. They were never intended to be a military or combat weapon, although I understand that the .357 Magnum was tested by the SAS at one time.
I see it being useful as a hunting handgun but I don't see any advantages over a revolver in that role.
The main Advantages over a revolver are reduced recoil (due mainly to the massive weight of the DE, and possibly a little bit to the gas operated action), and a different kind of grip shape than any revolver.
What you get is a pistol with a higher capacity than a revolver, shoots (some) of the same ammo as a magnum revolver, does not FEEL like a DA or SA revolver, has a slight increase in muzzle velocity, and a large decrease in felt recoil.
I got my first Desert Eagle in 1984, the .357Magnum (the only caliber at the time). Amazing piece. A tremendous blend of what was needed to run right, and "why the heck did they do THAT???" as a design.
I "discovered" the Desert Eagle well before Hollywood and the video gamers. Those who think it was just made to be a show piece are mistaken. That is what it got turned into, and the makers DID play to that market, once it became established, after all their job is to sell pistols!
I got my .44 Magnum Desert Eagle in 1990. I have no personal experience with the .50 cal version, so will not comment on that one.
The Desert Eagle is not quite like any other handgun. The people who expect them to be, often think the design is flawed, because it isn't exactly like every other handgun.
First off, they are very picky eaters. They are NOT ominvores. They require a specialized diet, and not every bullet and load are suitable. Some are expressly forbidden.
They require being held a certain way, or things often choke. One major point is the magazine. What I have found is that for best operation, the magazine must "float", meaning no pressure being put on it by the grip. If you "cup and saucer" the DE, it will often jam.
The most frequent complaints against the DE that I see are against its size, and weight. Personally, I have always thought that the designers went a little too far in this department, further than they
had to. A large size and weight is something you cannot get around firing rounds of the size the DE does, but I have other guns firing the same size rounds that are slightly smaller and lighter.
Which ever category they fall into, are there other guns in that same pot, or are they quite unique?
I have all of what I consider the "first generation" semi auto magnum pistols. Auto Mag, Desert Eagle, Wildey, LAR Grizzly, and Coonan.
.357 Magnum, .357AMP, .44 Magnum, .44AMP, and .45WinMag. I also have DA (S&W) and SA (Ruger) revolvers in magnum calibers. And a Contender with barrels for .357mag, .44Mag, and .45WinMag. I have a little bit of experience shooting them, as well.
Colt Govt Model and .44 Auto Mag
S&W M29 and .44 Auto Mag
.44 Magnum Desert Eagle and .44 Auto Mag
This should give you an idea of the size of the guns.
The Desert Eagle was not designed to be a badass movie or video game gun, the entertainment market did that. Can't fault the makers for taking advantage of it, though.
Hope this answers some of your questions, James. I have a lot more info I can share, if you want.