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No.
CVT’s only exist to be cheap. That is literally the only reason. Belts are cheaper than gears.
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No, they're not. They are quite pricy in fact, unless you're talking rubber belts in snowmobiles and ATVs/ SxS's. What they can do is produce best-in-class combinations of MPG and 0-60 times, which Nissan successfully did for a while there. 0-30 always sucked, but that is a weakness of the belt technology, not the principle of a CVT.
About 15 years ago, a 6 speed auto was in the $1- 1.5k range, the only 8-sp game around at the time was the premium ZF, coming in around $1300-1700, pricing for both dependent on models, and the Jatco belt CVTs started at $1500 and went up from there, LuK chain CVTs were even more. Understand that these prices are what OEMs paid the suppliers, not what you'd pay for a replacement.
To the OPs question, an unlocked torque converter is technically an IVT - Infinitely variable transmission - because it can have zero output speed with a torque reaction while maintaining a non-zero input speed, though unlike true IVTs, it can't seamlessly transition to reverse.