Kaiser definitely offers Medicare Advantage.
Your concept on Medicare Advantage is not quite right. All Medicare Advantage Part C are offered by Insurance Companies. They are not standing alone! Nor administrated by Medicare / SSA.
Someone said they also have Kaiser Medicare Advantage but their additional premium was about $20+. I replied that we asked our friend about it several years ago WHY his plan so expensive. He insisted his was Advantage but may be his was a PPO.