This chapter was really good. We get to see why Houjou exists, to demonstrate both Tae and Mai's growth since volume 1. (And also for the hilarity of the antagonist making the entirely valid point that it's weird for an adult to become friends with a 5th grader.)
Anyway. Were it not for Houjou, Tae -- or more realistically Mai let's be real -- could bullshit a story about her being Mai's aunt or something. With her there it becomes more complicated, but I'm sure she doesn't want the teacher to know why she's hanging around at school at night either.
By the way, Houjou's "you [presumably adults] are all the same" line, along with the fact that she is definitely the one playing the piano by herself, fooling around with anatomical models, and looking inside students' desks at night, strongly imply that she has severe family issues and is extremely lonely. I am curious why she latched onto Mai specifically, though. Maybe as a child who looks like an adult, she's a "safe" replacement for the actual adults in her life who are neglecting her?