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Gravity is why people lean into turns in the first place.
Those molecules have velocity that's taking them past whatever body they're using for a gravity assist. The gravity of that body grabs the vessel and begins to pull it in, being pulled closer to the body generating that acceleration and not simply passing by. The previous vector doesn't just disappear. All that matters still wants to continue in that direction of travel, including the people. The result is people having to lean into the arc of the assist because their own momentum is still changing to the new vector.
In short. Inertia is a bitch
I am not so sure.
In a gravity assist ( to be technical using a hyperbolic orbit of a large mass to change direction and gain velocity) the object is still following the curvature of space time. So the change in direction is affecting all particles in the object at the same time. There is no external force involve, so inertia does not come into play.
It is the same as people in the ISS orbiting the earth, they do not have to lean in the curve as they are following the curvature in spacetime around the planet. The only difference is one is a hyperbolic orbit (gravity assist) while the other is a ~~parabolic~~ elliptical orbit (ISS)
While the analogy the OP used of leaning into a turn is changing direction and there is a force that is only acting on part of the object (the tires of a car or motorbike).
Only one problem - the relative velocity of the ship is the same. Yes, the ship's vector got changed, but yours did too. Same reason you don't feel centripetal force in orbit, despite the constant vector change at 9.8m/s^2...
There is a miniscule amount of tidal forces felt during a gravity assist. Ultimately, the spacecraft and it's cargo are in freefall. The gravity assist would be barely noticeable. Remember that gravity isn't a "force" it's a bending of space changing the direction of a "straight " path. The spacecraft is traveling in a straight line with no other forces acting on it. The space that straight line is passing through is bent around the mass providing the gravity assist.