To me, fanfiction is about analyzing the source material and see what you can do with it to create something else while still paying the original work their due credits where possible, including keeping the characters in character. Just going by the simple "Ooh, I want myself inside there" "Wow I like this pairing" simply doesn't cut it, and is the cause of many a tale of horrid fanfictions we've seen thorough and thorough, not just in the FE fandom. I was guilty of this at one time, but took it upon myself to erase every single trace of that and reform.
Regarding "original characters" and "self-inserts," I do believe there are ways you can use them to help with the development of the story - as long as you don't make them too prominent (one of the symptoms of a Gary Stu/Mary Sue). A simple example is that you can write in first person, simply becoming a nameless observer and write about the characters and what they do, but in the mean time you don't say much about yourself. Canon characters may or may not notice you, but you make it a priority not to make yourself the focus of your own story, instead opting to tell your rendition of what you see in the characters and their interactions and actions with the world.
For all I know, we're writing fiction, not a scientific article or a university journal. Imagine if written fiction was done the other way round. I think we might be lost in trying to differentiate who the writer is referring to.