Maybe we should clarify "self-insert." The phrase has one meaning as "me but awesome," a device the author uses to vicariously live out their fantasy. Novels with self insert protagonists are usually tailored hyper-specifically to the author's taste, and have little appeal to most audiences. A self-insert protagonist in this sense is also one of several meanings of "Mary Sue." In this case the author typically will not use their own name, but someone who knows them personally might be able to identify a self insert easily. Self inserts are a great way to start writing, and most any author has written a self insert story somewhere early in their writing history.
A similar concept is the "author avatar," where the author inserts themself into the story, usually in a position where they have little to do with the plot overall. For example, author Paytoechip portrays himself as the god of chaos who governs the barriers between realms for all of his stories. I happen to quite like this kind of character.