The self isn’t non-existent per se. That would be nihilism. Buddhism denies both nihilism (nothing exists, has meaning or value) and essentialism (that things and people have an essence that make them what they are and that they have inherent/substantial meaning and value). The self exists in the same way all things exist: conventionally. Take a car, for instance. It has wheels, but they are not the car. It has doors, but they are not the car. It has an engine. It is not the car. You can do this with all parts. They are not car, and yet, if you remove all of them, no car is left. The same that is true of the car is true of all the parts. Also, every part is constantly changing and cannot be pinned down. It is not the same for two consecutive moments. Even parts that seem to be static are subject to microchange, which is why cars over time decay. The label car is a mental convention. A convenient way of referring to objects in everyday life. Saying that cars exist is not wrong. They do, conventionally. They just don’t exist “from their own side”, that is, there is no essence of “carness” radiating from the car, making it what it is.
In the same way there are thoughts, feelings and perceptions. We refer to things as “my” body, “my” thoughts, “my” feelings etc… What the “me” is changes, so we may say: “someone hit me” (referring to the physical body), or “someone hurt me” (referring to emotions) etc… The self cannot be pinned down in any of its constituent parts, just like the car cannot be pinned down by pointing to any part of it that supposedly is the car itself.
There is also the added fact that perceptions, sensations, thoughts, feelings etc… are all appearences to awareness. You know this screen because of awareness of it. You know your thoughts because of awareness of them (and without any object, no awareness would exist. It has also co-dependently arisen). Why call one “self” and the screen “other”? It is one field of awareness where both the body/mind and objects are bundles of sensations, perceptions, thoughts, feelings etc…