If you read the details of the Joule experiment you will find that the weight fell at essentially constant velocity. That would mean no change in kinetic energy of the mass from the beginning to the end of the fall. The increase in kinetic energy to get the weight started equaled the decrease in kinetic energy to bring it to a stop. The only change in energy of the falling mass was a change in potential energy. Therefore $W=mgh$.
However Kinetic energy is not equal to 0 so 𝑊≠𝑚𝑔𝐻
It is the change in kinetic energy that is equal to zero, not the kinetic energy itself. If the change in kinetic energy from start to finish is zero, there is no gain in kinetic energy that needs to be accounted for. Only the change in potential energy needs to be accounted for. And that is accounted for by the paddle wheel work.
You are probably aware that if you lift a mass off the ground starting at rest, raise it at constant velocity and bring it to rest at a height $h$ there is no change in kinetic energy. Therefore the positive work you did in raising the mass is $mgh$. At the same time gravity did negative work taking the same energy you gave the mass and storing it as gravitational potential energy. The net work done on the mass is zero by the work energy principle which states that "the net work done on an object equals its change in kinetic energy".
The Joule experiment reverses this process. Gravity does positive work in lowering the mass but does so at constant velocity so there is no change in kinetic energy,
only a change in gravitational potential energy. Therefore the work done by gravity equals the paddle wheel work
I think you may be confusing what is going on in the Joule experiment with what would happen if the mass were in free fall. In free fall the mass accelerates increasing its velocity on the way down and therefore increases its kinetic energy. According to the description of the Joule experiment, however, the velocity of the descent of the weight was controlled to be constant. Therefore there was no net change in the kinetic energy of the mass.
Hope this helps