On my side, I will write with respect to the comics. In particular, I base myself on Journey into Mystery #622 to #645 (the whole Kid Loki arc, from Fear Itself to Everything Burns) and the 2014-2015 Loki: Agent of Asgard (the current series that tell Loki's current adventures).
First, let us get that out of the way: the comic-book Marvel universe is not particularly consistent - I am sure there are authors that treat Asgardians simply as advanced aliens, as the MCU does.
However, there is another interpretation that is pretty prevalent in recent stories, that is summed up by Journey into Mystery as:
Gods do not have history. They have story.
That is, the gods of Asgard, of Otherworld, and of the Olympus are not simply extraterrestrial beings that live in another dimension, they are also made of stories. The examples of that are multiple:
- In Journey into Mystery (Fear Itself), Loki weakens the Serpent by rewriting his story;
- In Journey into Mystery #645, it is revealed that
Loki wrote Hela into existence, and made her exist an eternity before he did;
- In Loki: Agent of Asgard, Loki carries his missions so that his previous crimes can be erased from Asgard's histories - the thinking being, his crimes being forgotten is the same thing as these crimes never having happened in the first place!
So, why is that whole explanation salient to the question? Because if Gods do not have history, but story, then in a real sense, gods exist only because of, or as part of, the stories that are told of them.
In other words, the stories told of the Asgardian gods by Norse people are not a product of the existence of the gods - the existence of the gods, and their stories, are one and the same.
If you argue that it does not make sense logically, I agree. But stories do not have to. (See also: magic)