I mean, I like it as a cute comedy love story, but it's really not good at being a supernatural drama as well, despite trying so hard.
I don't think it's trying to be that at all. The sheer dismissive all characters immediately show at any sort of problem suggests to me this is done on purpose (and I mean, same author as Prunus Girl, they didn't just forget how to write "uncertainty" ... )
I don't even think that the points brought up need to change much (although that might not hurt either). But I do agree that there should be something to frame the events so far. Doesn't have to be relationship drama or even the typical sinister government organization or secret Vatican vampire hunters.
But, for example, there could be something about her dealing with the fact that everyone (except Nana) will die long before her. And it could be expanded on her figuring out what she wants to do with her immortality etc. That wouldn't even take anything away from the slice-of-life-ishness of the series.
Or maybe something about other vampires, or maybe there are other supernatural creatures around that can add a bit of "let's explore this new supernatural world I'm now part of" or whatever.
I do like the series - and the absence of a story is kind of part of that - but it might not have hurt having one either.