It would work, but as Allen noted, you may have some serious chatter with this type of tool. One way to work around this is to bore close to your finish depth, then feed in the last few thou into the work while rotating the lathe chuck by hand. This allows you to scrape the last bit of material off of the cut surface. If all goes well, the chatter will be removed and the diameter will be fairly true.
The key here is rigidity of the machine, setup, tool and workpiece. Also, the tailstock has to be lined up carefully with the spindle, and the cutting tool has to be made so that the diameter formed by the cutting edges is concentric with the shank.
For the tool, it would be better to make it out of an old cut off large diameter Morse taper drill shank to fit directly into your tailstock. It would probably be too flimsy (more prone to chatter) if you held it in a chuck in the tailstock.
Also, a less chatter prone tool design would be one with four, rather than two cutting edges. This could be made similar to the one you show, but with a 2nd slot cut through at 90 degrees to the 1st. Two more cutting inserts, ~1/4 of a circle each could be added perpendicular to the 1/2 circle one. Silver solder the parts together and then finish grind the cutting edges.
One caveat here; there will be a large web area that will have to be relieved so that the tool will cut in the center. One workaround for this is to make the two extra inserts with a flat ground at the nose so only the main (1/2 circle) insert is doing the cutting at the very center of the tool.
I have a tool like the above somewhere around here.