> Everything you gain from gravity you have to give back on the way out.
This is true only in the center-of-mass frame, and that's true for all conservative forces.
If I have a bowling ball covered in springs and I throw it at a marble at speed v, the marble will traveling at a speed 2v after the collision. It would be completely correct and not misleading to say "the springs on the bowling ball accelerated the marble to 2v" even though the marble's speed in the center-of-mass frame is the same before and after the collision.
It's also true that the energy gained by the marble comes from the bowling ball in the rest frame. That doesn't make the first statement wrong or misleading. It just means that you like thinking about things in the center-of-mass frame.
> would be completely correct and not misleading to say "the springs on the bowling ball accelerated the marble to 2v"
I suppose this is where I disagree. The springs transferred momentum. Your throw (and the resulting motion of the bowling ball) did the work. (Tyres don't accelerate a car, its engine does.)
You could slingshot around a moving body magnetically, the fundamental principles remain the same. The effect of a gravity assist comes from the motion of the body, not gravity per se.
This is true only in the center-of-mass frame, and that's true for all conservative forces.
If I have a bowling ball covered in springs and I throw it at a marble at speed v, the marble will traveling at a speed 2v after the collision. It would be completely correct and not misleading to say "the springs on the bowling ball accelerated the marble to 2v" even though the marble's speed in the center-of-mass frame is the same before and after the collision.
It's also true that the energy gained by the marble comes from the bowling ball in the rest frame. That doesn't make the first statement wrong or misleading. It just means that you like thinking about things in the center-of-mass frame.