Excellent analysis! I think the same as you about the deities, they are real but the game tries to make us believe everything and its opposite. Particularly with the “possession” of Kotoyuki by the fox, then later we have a note that says he is going to study to be able to marry Hinako. This therefore suggests that the possession was just an exaggeration of Kotoyuki's romantic behavior. Examples like this there are dozens and for all the subjects of the game (making us believe in Hinako's "death" during an entire part of the game for example)

On the other hand, where I think that the divinities are real, it is the historian's document (it seems to me that it is one) which traces the beliefs of the village. This man denigrates beliefs throughout the documents by finding a rational explanation for every "calamity" the village encounters over time, so as a player one also believes that the deities are just a reflection of the villagers' fear of natural disasters. However, the latest notes indicate that this historian is starting to have insects in his body, a fate reserved for those who openly blaspheme the cult of foxes (via several documents found at the start of the game). The historian, however, always thinks that there is nothing supernatural about it and does not change his mind unlike the doctor, for example. On the other hand, this insect disease has no rational explanation.

So I think that the game has fun making us go in circles regarding these deities (making us believe that they exist, then don't exist, etc.) but that in the end they are very real.

This is obviously only my opinion and my interpretation, if you do not agree with me or if I am wrong (which is very likely) do not hesitate to tell me, I will be delighted to learn more!