With gas sitting in a carb for 3 years, it will take quite an effort to get it thoroughly cleaned out. Since you say it won't idle, I'd say you'd best doublecheck that the pilot jet isn't clogged. You really shouldn't have to replace the entire carb, though.
With regards to the CVT, it usually is the superior choice for a yard kart. However, I must ask- is it a live axle, or dead? The difference is whether the engine drives one rear wheel, or both. Converting a dead axle (1wd) kart for use with a CVT can prove to be a bit more challenging with the clearances involved. In addition, all the extra torque from the CVT can create a lot of extra wheelspin when accelerating from a standstill.