Homemade lathe cutting tool from broken drill bits - video

[Toolmaker51]

Drill bits, sure. And any other cutting tool...tap, end mill, center drill, reamer etc., as long they exceed file hardness. Using lathe as a grinding fixture is acceptable. A low spindle speed and correct feed makes this go quick. Find sweet spot when metal removal isn't burning part or degrading wheel quickly.
If using spindle create diameter, the best result is when grind wheel and part counter rotate. Just like conventional vs climb milling.The little wheel shafts won't dig in much, but deflect away badly; climbing more material than they can digest. Doesn't matter which is clockwise, but grind motors aren't normally reversible.
BUT;
I'll recommend cloth is not adequate protection for ways; abrasive grains and swarf are smaller than most cloth or paper towel.
Use plastic food wrap, cooking parchment or foil, oiled newsprint, or packaging cardboard [crackers etc] also oiled.
The oil makes swarf and abrasive adhere while working, and more important when removing the shield.
The bedway thanks you, and the wipers won't snarl.

Control carriage travel to a stop, any kind of bumper just clamped on, will protect grinding wheel, and tool bit as well. Smaller they are, easier to spoil.