mild cases of appendicitis can be treated with antibiotics, but depending on the health systems surgery is still the standard of care, however it doesn't generally present itself with this kind of pain, but it's true that everyone pain tolerance is different, so hard to actually judge.
Also the term for non immediately life threatening (aka not at immediate or short term rupture risk or septic shock, but can still be threatening if left unchecked and if untreated can develop to a point it require surgery) is Uncomplicated Appendicitis and can be treated with antibiotics. Uncomplicated however doesn't means necessarily mild, and more severe cases should still be actively monitored.
Considering the dialogues I suspect the author confused mild with uncomplicated, as, while severity doesn't actually seem high, it may be a medium severity case that started as mild.
Also oral antibiotics (that don't require hospitalization) can be used with similar (only a bit lower, 70% vs 74% (for oral + IV) for 1 year recurrence free) efficacy, however it's an overall recently studied protocol, and japan health system (but not only japan) is somewhat know for therapeutic inertia. Of course for uncomplicated but high-severe cases IV may still be preferred for speed of action.