G'day MF
A great question to which I will add my 2-cents.
The continuously variable transmission comes in a number of forms.
I guess, just like gear boxes, there are good and bad ones.

https://en.wikipedia.org/wiki/Continuously_variable_transmission

For example, the CVT on my MP400 motor trike is total rubbish.
It is harsh on take-off and in bumper-to-bumper traffic.

In a Toyota Corolla Hybrid it works like a good automatic.

In all cases, the CVT should deliver best fuel economy -
provided that criteria was part of the design...

But, as you say, the issues are more complex.

The CVT on vehicles is older than folks think.
I remember an old doco that covered this odd vehicle...
{I think one of these crossed AUS East-West ...

https://en.wikipedia.org/wiki/Variomatic

If you like driving - CVTs are not the best choice.
They average out to optimize energy consumption and performance.
Manuals enable choice for the Dr Jekyls and Mr Hydes amongst us.

Cheers
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Jack