In a two-body system it doesn’t matter what you do; gravity is a conservative force so conservation of energy demand that you leave the body’s SoI at the same speed you entered it (in the body’s frame of reference).

You can lose speed or alter course relative to another body in a single encounter, and those changes can reduce speed in future encounters, but if you’re on an escape trajectory heading in you stay in one (without forces beside two-body gravity, which is a pretty safe assumption 11 AU out of Saturn doesn’t come close).

> In a two-body system it doesn’t matter what you do;

Two-body systems do not exist in reality.

Energy is also conserved in 3 body problems. When you utilize the slingshot effect, some of the energy of the orbit of the body you are swinging around orbiting is transfered to you. The transfer of this energy does not depend on the closeness of the sun, but rather on how deeply you descend into the gravity well of the object you are slingshotting around.

> which is a pretty safe assumption 11 AU out of Saturn doesn’t come close

No, it really isn't. The "safeness" of the assumption entirely depends on your margin for error. The existence of the naturally captured saturnian satellites clearly indicates that you are simply wrong about the relevant margins for error.

Saturn's captured satellites might also be the result of incidental aerobraking or whatever you want to call smashing into a bunch of very tiny satellites during a close periapsis, no?

You need some sort of subsequent acceleration to raise the perigree out of the atmosphere so the orbit doesn't continue to decay. This could happen due to a slingshot effect, but atmospheric braking alone is not enough to allow you to establish a stable orbit.