when I broke my wrist and had to be casted all until the elbow, the next day I was in school like nothing, granted I did techincally have an exception on writing (it was my dominant hand). Now I realize a broken leg is worse for going to places, but unless the school is far and there are no public trasportations (and even that they could handle with the parents help for the month) there is not impossibility, there are special devices to assist mobility like crutches. I did see people mounting with crutches on public trasportation in my city, sometimes with a casted leg, going to school, university or somewhere else. When a high school class member got an encounter with bus-kun, and got quite hurt on multiple part of the body, he did skip school maybe a week. skipping school (not really as she still had to follow classes remotely, also a remote less cannot be compared to a in presence lesson, if the teacher need to follow both in presence and remote students, remote students will have less attention from the teacher) for a month for a broken leg would be outrageous, unless there are extremely peculiar conditions. The author didn't even show a proper cast.
Also I'm not sure how it works in Japan, but in my country a doctor cannot put you in sick leave or ordering you to stay home in most cases. Or better he "can" order you to stay at home, but you are not bound to follow it, unless it's aninfective issue with an active quarantine protocol, like Covid19 ones(and there was a decree for this specific case that forced the doctor to put a worker in sick leave, make sense for in presence worker but stupid for remote workers).
Another institute is the requiriment of medical certificate for illness related absences superior to 3 or 5 days (depending on region and school grade), that depending on the specific regional regulations at specific times, or their interpretation, could include "non school days" in the absence count, so missing first day of school of the year could trigger the requirement. Luckly (becouse it's totally useless) many regions in recent times removed the obligation(could have been meaningful to only restrict it for infective illnesses, to allow the doctor to check for more hidden symptoms that may indicate an infections still active, but removing the absence requirements), and other regions allow for self-certifications for absence not related to illnesses (and accidents can probably be listed as not ilnesses).