Unfortunately batteries are definitely not improving exponentially, and they are limited by chemistry which has been kind of a known quantity for a very long time. There will be progress but it will be very slow. A bit like 100UL 
The main progress is in re-packaging the “battery container” (the car) so it can hold more of them 
Indeed, battery technology is plateauing at the moment. It would need some fundamentally different chemistry to advance the 5 – 10 x energy density that is required to make it work.
A major part of the ‘re-packaging’ for cars is extensive use of aluminium and lightweight composites in a structure that was previously mostly steel, which helps offset the weight penalty of carrying batteries and is part of the reason EVs are so expensive.
Not so much an option for aircraft, which already push the lightweight structure policy to practical limits.
Then you have two other aircraft-specific issues – firstly the ability to trade payload for range which is rather difficult with batteries, and secondly a lot of long range flight leans heavily on the fact that the aircraft gets lighter as fuel is consumed, again rather difficult with batteries.
Peter wrote:
Unfortunately batteries are definitely not improving exponentially
There is definite evidence that batteries at least used to improve exponentially in terms of kg/kWh such as eg: https://web.mit.edu/cmagee/www/documents/50-onlineperformancetrendsfirstpublication.pdf
Also more recent evidence that they are still improving from at least 2021 (but not sure if exponentially). I wouldn’t even be worried by a plateau because in other fields we have also seen the phenomenon of exponential growth → plateau → paradigm shift (I guess different chemistry in this case) → further exponential growth.
Graham wrote:
Then you have two other aircraft-specific issues – firstly the ability to trade payload for range which is rather difficult with batteries, and secondly a lot of long range flight leans heavily on the fact that the aircraft gets lighter as fuel is consumed, again rather difficult with batteries.
At least it simplifies the W&B calculations 
That is some academic paper from 2016.
Re W&B, you still have to get off the ground 
Heart Aerospace is making progress!