One often overlooked reason to shoot raw: smart objects.
Process the image in raw to get the main layer/subject as best you can. Open as a smart object. Make a new smart object from that layer. Open the new object in raw and you can basically start over again, adjusting specific areas of the image. Using layer masks and the paintbrush tool you can make multiple smart object layers and open each in raw to fine tune the image so that each section of the photo gets exactly the treatment it needs.
I don't understand how this is something specific to RAW. It sounds like you're describing photoshop. Am I missing something?
Smart objects open automatically in RAW. So, each smart object layer gives you the full range of RAW adjustments. Basically, you can create an unlimited number of layers and fine tune each one in RAW and then pick and choose the parts you want. And, since each layer is basically a RAW file, you can go back at any time and fine tune any layer.
You are correct in that you can certainly use layer masks and adjustment layers to accomplish some of the same things in Photoshop, but the range is much more limited. For example, you can create an exposure adjustment layer or a contrast adjustment layer or a saturation adjustment layer, but you cannot create a layer that adjusts all three in one.
With a smart object, you can create a layer that will open in RAW, allowing you to adjust exposure, saturation, blacks, contrast, whites, etc., in an infinite number of combinations all on that one layer. Then, when you return to Photoshop, you can use a layer mask to reveal or conceal the adjustments on that layer.
Take the dog example above. To my eye, the sky is still a bit overexposed and the dog's face is still a bit underexposed. I could take one smart object layer and in RAW adjust exposure, contrast, blacks, highlights, etc. to get the sky exactly as I want it. Then, I could make another smart object layer and adjust the dog's face to get it exactly as I wanted, again making multiple adjustment in RAW to that one layer. Then, it's just a matter of adding a layer mask in Photoshop and painting out portions of one layer to reveal the other.
Could I do that in Photoshop alone? Well, possibly, but I just wouldn't have nearly as many adjustment tools available. I could use an exposure adjustment layer on the dog's face, but usually an exposure adjustment alone won't do the trick. In RAW I can tweak the face so that it has a bit more exposure, but still preserve the deep blacks, protect the highlights and keep good contrast in the face.