So, to the question: “with what key levers to start a basic sell-out analysis?” I would answer: “with the one that seems to be most relevant for the category and the channel in question”.
Perhaps it is an answer that helps very little, but it is the most logical. For a manufacturer of “liquid” products such as milk, water or beer that have as characteristics large pack sizes and high levels of rotations leading to in-store replenishment problems, reducing out of stocks will clearly be a priority. On the other hand, for those who are dedicated to manufacturing chocolate bars, a category where fierce competition is focused on the field of promotions, it will be necessary to optimize above all promotional mechanisms and visibility.