I can understand your affinity for autos. I prefer them as well. I've tried to like several revolvers, but they just feel weird to me.
The Desert Eagle makes a fine hunting handgun. While I can't speak for its accuracy compared to revolvers, as I have never shot nor seen them compared, I've seen my dad shoot a hanging steel tie-plate at 100 yards from the supported offhand position 6 out of 8 times with iron sights and his Mk VII .44 Magnum Desert Eagle.
The Desert Eagle is gas operated and fires from a rotating bolt. This makes the barrel stationary, unlike most handguns that operate on the modified Browning mechanism. So while it may or may not ever be as accurate as a tuned revolver, it is among the more accurate autos on the market, and more than adequately accurate for hunting purposes.
Also, the new ones, the Mk XIXs, come with scope mounting rails on their barrels. This makes it easy enough to attach a red dot or EER handgun scope if you so desire.
The .44 Mag is plenty for deer. A good 240 gr JHP at ~1400 fps will put any deer on the planet down in short order if delivered to the boiler room. The .44 Magnum Desert Eagle is a soft shooter that is much easier to control and shoot accurately, and tends to be more reliable as well. The Desert Eagle is reliable, but susceptible to limp wrist failures. These can be more common in the .50 AE because of the added torque provided by the bigger bullets.
While we're on the subject, the Desert Eagle also requires jacketed bullets as hard cast can clog the gas port. Most common bullet weights and profiles will function fine, the Desert Eagle won't offer the versatility of a revolver. It simply won't handle the range of loads a revolver will. Heavier bullets with large flat meplats can get hung up on the feed ramp of the Desert Eagle, and it can't shoot "Special" loads, like .44 Specials. It is a large handgun that can give people with small hangs trouble, and it will be heavy as well.
The weight helps control the recoil, however, and isn't bad in a good holster. It is accurate, reliable, and easy to maintain. And it is incredibly robust. My grandpa put a very overloaded .44 Mag round through his. Not only did the pistol protect him from injury, but it remained relatively undamaged itself, and actually functioned. He just took it to the gunsmith who cleaned molten brass from the gas port and firing pin channel, and replaced the extractor and ejector. The pistol continues to function fine with no apparent ill-effects from what we later calculated on Quickload to be an estimated 60,000+ PSI.
The .50 is a hoot to shoot. Those big 300+ gr bullets land with a satisfying thump. After shooting the two side-by-side, there can be no question that the .50 is more powerful. I doubt the differences will be appreciable on deer, and the .50 is more expensive.
I say get the .44. If the Desert Eagle meets your requirements, you can add a .50 AE barrel later at your discretion. Swapping between the two calibers is as simple as field stripping it and slapping the other barrel in, then reassembling and loading with mags for the other caliber--barrel and mags, that's it. It takes less than 30 seconds.
.44 Mag on left, .50 AE on right.
As for the Desert Eagle--I like it. But it does have limitations. If you accept them, you and your Desert Eagle can have a long and fruitful relationship. If not, your Desert Eagle will end up a safe queen.