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A local seller cut down an enormous Euphorbia Ammak Variegata and sold off some of the large arms as cuttings. I have two of those pieces, and when inspecting just now I see that they are at the very first stage of developing roots. The plants have been almost a year now trying to root, and they have become extremely emaciated. Does it ever make sense to just leave the roots in a plate of water and let it drink for 24 hours to try to rehydrate the euphorbia? Photos showing the plant at both ends are attached.

It's better to pot it up and then water it. If you keep it in a bowl of water the roots that grow will be weak and soon as you transfer it to soil thdy die. (keep it in a bit of a shady area also.)

In addition to what Russell99 says, the roots are tiny so won't be able to absorb much water anyway, and some Euphorbias will rot if you try to root
What you will probably find is that now it's started to root, root growth will be quick, so use a very porous soil mix and water it lightly to encourage this. If you leave it somewhere warm and in the shade, you may notice it filling out in 2 or 3 weeks anyway.

I'm going to share my experience with rooting euphorbia. I have 3 that I acquired as cuttings. I did a fresh cutting on all three. Let them callus over for a week. The first one did nothing for 4 months. Then it started to rot so I cut it back again. Replanted it in fresh soil and magically 2 months later it rooted. The second did nothing for 3 months. It rotted so I cut it back. It rotted again, so one more attempt at cutting back. Fresh soil each time and after 3 months it rooted. The third never rooted and died off.
Replant yours in fresh dry soil. Since it has grown some roots, give it a week then give it a light watering. You'll know its going to grow when you see signs of new growth at the top of the plant.

I was not proposing to leave the plant indefinitely in water. I was just wondering if a short stay in water would allow the plant to rehydrate its internal tissues faster. That does not imply enough time in water to grow roots.

@theclosetguy I have also found Euphorbias are very prone to rotting. I find that if I leave them out in winter rain that I must use a pure inorganic soil like pumice. My Euphorbia White Ghost were in "the gritty mix" (equal parts of sized bark, stone, and surface). They thrived in that in summer and rotted away as soon as winter approached. Gritty mix is very porous so that was an eye-opener. My Euphorbia Firesticks, on the other hand, seem to just look malnourished in pure pumice and want something richer. I find it really hard to keep Firesticks alive in my zone 9a outdoor environment. They take off in summer and rot in winter, when left outdoors.

I have this Cleistocactus strausii (silver torch) that is at least 18 years old. It started out as one stalk. It broke. An offshoot started. then another offshoot until I had 4. 2 off those broke off and I was able to get one to root. The other just rotted away. From that one rooted offshoot, 2 more grew. We tried to move it one day and 2 off the stems broke off. I was able to get one to root. More offshoots/stems have grown. 4 weeks ago I had to move the plant 2 feet. 5 stems broke off. The skinniest rooted and started to flower. The other 4 refuse. All were about 2 ft long. I let them sit and callus over for a week. Put them in fresh dry soil and 1 week later they fall over and I have to remove 6 inches of rotted plant to get to clean tissue. This has happened twice now. Now they are 12"long and I'm letting them callus over again. I'm going to give a few extra days this time.
Who knows why some cactus cuttings will root and some don't.