Smooth ends of steel wire after cutting it?
Your objective is much easier than you might expect. I frequently use a small belt sander or my Dremel rotary tool to round sharp edges of cylindrical items such as nails or stiff wire.

Rather than using a grinding stone on your Dremel, consider to use a sanding drum. The grit will be more coarse on the drum and remove metal faster. Unless you require extremely smooth finish, the drum will make for shorter work.

Regardless of the tool selected, you'll want to secure the Dremel to a solid workspace/workbench to provide for best control. I've done the process with a Dremel rotary tool in one hand and the workpiece in the other, with mixed results.

Once secured and powered up, you can use both hands to position the wire to the drum/stone and brace one hand for graduated movement. You won't want to push the wire into the tool from an inadvertent muscle twitch or a slight balance waver. If you can perform the task while sitting, all the better.

You mention "properly smoothing" which means you will likely aim to the grinding stone side of things. The amount of material to be removed is small enough to make this practical.

Once you've attained the desired shape, your Dremel tool may also have a buffing wheel of sorts. It's likely to be a white fabric looking disk with a mandrel or a center hole for a thread-in mandrel. Using this tool in the same manner as described above, after applying jeweler's rouge, will provide for a polished finish. When performing the final polishing, use caution to avoid creating so much heat as to burn your fingers.

Spin the wire if possible as you press it against the wheel. The wire will accumulate abrasive powder as well as some of the fabric from the wheel, requiring periodic wiping to better examine the surface.