This is not a serious response to this question, but last night I was thinking about this question before I went to bed and I had a dream that there was a fifth Hogwarts called the "House of Gingerbread" ??? Unfortunately I remember nothing else of this strange, cookie-themed Hogwarts house.

ANYWAY. For the question, I'm not really sure if 4 houses is necessarily enough to sort EVERYONE. But I think that maybe having fewer options is better because that way you don't really have to cover all of the bases as there will be more variation within the houses as a consequence. Whereas if you had 12 different kinds of houses, you open up less wiggle room in each, you know? I'm also kind of imagining, like, what if students were sorted based on their Myers-Briggs types. There's obviously variation within each type, but holy cow if I was in a house with all other ENFJs I might blow up at some point because being surrounded by people who think / express themselves like me 24/7 might drive me up the wall.

I've been thinking about the Hogwarts Houses vs Ilvermorny houses though and find that I strangely prefer the way that Ilvermorny breaks up the sorting, but I think that might just be because I identify more with my ilvermorny house than I do with my hogwarts one. I'm still thinking about what my potential solution / fifth hogwarts house may be though. I'll try to think on that some more.