Very well said. I imagine the reason why the self-insert things come rather often especially around Nodoka is exactly because her own story seems to be going nowhere, or hasn't started like you said. Thus when thinking about the in-story purpose of smartass characters such as herself, people have no other option but default to assuming their roles being either plot device or creator's self-insert. Neither of those is in-universe and thus neither of those is a good trait one assigns to characters that player can interact with on a daily basis.
The most generous benefit of the doubt I can give regarding Nodoka's purpose is that perhaps she is there mainly for tapping into cosmic secrets, and nothing more. Aka, understanding her episodes, her limitless knowledge, the upside down house, Kyoko Nagasawa, and understanding this senseless world through or with her are her purposes; and the rest will either be plot devices for others or simply go nowhere.
Like, at this point of time I've almost completely given up on Nodoka route receiving any interpersonal development content. Judging from how her book arc concluded and how things just seamlessly transitioned to Nodokathon, I sincerely don't know if she'd ever get her fair share of a long overdue kick in the shin. Maybe she will in the future, but after waiting months for any fallout to happen and none happened, I might as well believe that Selly personally tailors and puts a full set plot armor on her, grovels at her feet, before going out to get her coffee.