Regarding the last part of your question, the thing that you are neglecting is the part of equal and opposite reaction. Suppose I am standing in a train carriage and throw a ball. Then not only do I accelerate the ball forward, but I also accelerate the train carriage backwards by a very tiny amount. In a frame where the train is standing still, this means that I use chemical energy both to impart kinetic energy mostly onto the ball but also a tiny bit of int onto the train.
Now do the same thing in a freely rolling, moving train (or a respective frame), such that the train is moving in direction throw (respectively the frame is moving in opposite direction). Then on one hand, the ball gains more kinetic energy, but on the other hand, the speed and thus the kinetic energy of the train now gets reduced. So what happens is that you use chemical energy to impart kinetic energy on the ball, but you also convert some of the kinetic energy of the train into kinetic energy of the ball. If you take that into account, you will find that the chemical energy needed in both cases is the same. (As will be the case for any potential energy if defined properly, only the kinetic part changes.)
In case you don't want to be on the train, you can also do the same thing standing on the ground. If you throw a ball, then not only are there forces between you and the ball but also between you and the ground. In a non-moving frame, the latter perform no work, as they don't cover any distance. In a moving frame however, the same force covers a distance and thus in some sense the ground performs work, which again precisely produces the missing energy needed for the change in kinetic energy of the ball.
This is irrespective of the source of the force, even if it was not a contact force like a gravitational or electric field. At the end of the day, these fields have a source! If this source was taken into consideration, we would get the same work done.(Btw, contact force is electromagnetic, so it's no different)
Then the problem was raised from:
- attaching the frame to one of the interacting bodies and this made it non-inertial.
- asking about the work on the ball and not noticing there are other works as well that should be taken into consideration.
What this says really is that you need to take into account all the interacting bodies so that all forces are internal, and conservation of momentum applies, and if this is the case, the change in total kinetic energy would be invariant and total work would be invariant as well.