Whatever speed an objet may gain while entering orbit should be lost when exiting it, right ?

That is true from the frame of reference of the planet. From the frame of a 3rd distant object that you want to accelerate towards, it appears you have gained momentum.

So I guess it’s the cinetic energy of the planet that is actually fuelling the spacecraft, isn’t it ?

Yes, but the mechanism for 'extracting' the kinetic energy from the planet is by using 'gravity', hence the name, "Gravitational Slingshot".