That's only thinking about it from the perspective of the planet. Relative to the planet, the magnitude of the incoming velocity vector is equal to the magnitud

That's only thinking about it from the perspective of the planet. Relative to the planet, the magnitude of the incoming velocity vector is equal to the magnitude of the outgoing velocity vector. But in the reference frame of the sun, the planet loses an infinitesimal bit of momentum, and the spacecraft gains that momentum. And that's assuming you don't do an extra burn at periapsis, which is far more efficient at changing speed than doing the burn in interplanetary space.