Unlikely that the stains will be completely removed, even using a poultice, but that may not matter. Cleaning and conditioning will darken the leather enough so the stains won't be as noticeable when finished. Uneven appearance is part of the history of aniline leather furniture, call it patina if it makes the owner feel better.
The best leather has to be used for aniline pieces and any leather can be "protected" so if you paint this it loses that special quality. Paint analines only as a last resort.
That's correct Oz, that's why I suggested the painting purely as a last resort.
This does firm up the leather and removes the luxurious feel, or hand.
On cushions, if one can get to the inside and draw the oils to the back of the leather, you have a better chance of restoration.
The other option would be to replace the damaged sections, finding a suitable Aniline color that matches could be a problem.
For those unaware, Aniline's get their color in a dye bath whereas Pigment/Protected leathers are sprayed with a color, even though most also go through a dye bath.
