What you seem not to be acknowledging is that velocity always has to be measured relative to something. If you only consider 2 bodies (planet and satellite), the incoming and exit velocity of the satellite relative to the planet will be the same, as Dave originally said. However the velocity of the satellite relative to a 3rd body, say the sun, can be altered, as a result of the planet being in motion relative to the sun.
For observational and practical purposes I can see this, but is a third body absolutely necessary for a body to be in motion, independent of observation?
The orbiting planets are handy because we know their precise behaviors and can use that knowledge to calculate trajectories, etc.
But as I understand it as long as a massive body is in motion (momentum) it can be used to increase the speed of a bypassing spacecraft through the slingshot effect (gravitational assist)
Is the illustration above incorrect? It merely shows a massive body in motion and a spacecraft gaining speed from the gravitational assist of that body. A two body interaction.
Does relativity to a third body have anything to do with this physical phenomenon?
At its fundamental level it comes down to some sum of motions and gravitational pull of the massive body and the motion of the spacecraft using that gravitational pull, no?
The third body (like our sun) is what keeps planets in orbit, but that just means that the orbiting planets possess motion (momentum) relative to the sun, which is useful, but not absolutely necessary. Only motion and gravitational pull between two bodies are necessary for a spacecraft to alter course or increase speed. (as illustrated)
The underlying physics is not concerned how or why a massive body has acquired its motion, as long as it is in motion and if a spacecraft can plot an in-flight precise intercepting trajectory, it should be able to use the gravity pull of that body, for use of altering course or even increase it's speed from the gravitational assist.
The gravity of the third body (the sun) establishes predictable planetary orbits, but the slingshot effect has nothing to do with orbits themselves, but the speed and momentum and gravitational effect of a massive body are key, regardless if it is in orbit within the solar system or follows a straight path outside the solar system.
Does this make sense?