First time poster here. I've found this to be an interesting place! I haven't been around long enough to know if DaveWing is serious, but if so, here is my attempt at simplifying it. As an aside, I have spent most of my career doing analog/power/battery/charger designs in a position where I am required to assist/teach design engineers with a wide range of experience from rookies to the most Sr. Engineers. I've learned patience
In your example, let's make the following assumptions (you can change any of them and rework the math, it doesn't change the results, this just makes the numbers work out simply in your head).
1. Let's ignore parasitics (battery ESRs, charging efficiency, etc.)
2. Your "source" batteries on the left are 1.5V each.
3. Your "charging" battery is at 1.5V
This means 1.5V across the motor (3V -1.5V). Let's say that with the mechanical load and 1.5V across the motor, the current is 1A. This means 1A through everything in the loop.
So, you are sourcing 3V*1A=3W. The motor is using 1.5V@1A=1.5W and the "charging" battery is also using/accepting 1.5V@1A=1.5W. You are sourcing 3W, and using 3W. What is the mystery?