The misunderstanding is that she thinks the displays of politeness are evidence the people care about her in a deep and long lasting way. But they don't, they're simply being polite, and when they no longer want her services they don't want to interact any more.
You’re assuming that in the experiences she had with those students, it was their politeness that she interpreted as friendship. It’s also possible that she can discern between politeness and friendship and was seeing genuine friendship.
Empirical observation #1: A student said “I’m sick today. I can’t make the lesson. I’ll contact you later,”
Empirical observation #2: That student didn't contact her later including not replying to emails she sent afterwards.
There's also a direct consequence of empirical observation #2: The lessons could not continue.
Assumption #1: The student pretended to be sick. This student could have been actually sick. We have no evidence either way.
Assumption #2: Your definition of "genuine friend" is the same as the teacher's definition and the student's.
Assumption #3: Using the term "obviously" shows that you've assumed that I (or possibly anyone reading this) have the same cultural values and definitions needed to draw the same conclusion that you have.
I know I'm being pedantic and apologies if my tone comes across as condescending. I would normally avoid replying in these situations but when I put myself in your place I appreciate having the other person's perspective even if I don't agree with them because it helps me to clarify my communication in future conversations.
You can have as low a bar for "genuine friend" as you like, and I don't think the final evidence (one step text dismissal) can clear it.
That said the author maybe does see the fakeness ("They want it to feel like a friendship" ie. not be an actual friendship, just something like that feels like it, fine line I know but it's desired performance vs actual emotion). However she still thinks they "owe" her something.
But English lessons in Japan must be like... I dunno, grocery shopping to us. I'm guessing tutoring is more common in Japan than in North America. When you decide never to go to a particular grocery store, you don't give long heartfelt goodbyes. You just stop going.
It's not my assumption, I'm merely going by the author's empirical observation. No one whose only real connection to you is English lessons that pretends to be sick one day to cancel their lessons forever is a genuine friend, obviously.