Months are one of the best ways for a low-tech/pre-tech culture to keep track of dates (using the Zodiac for something it can actually do—act as a calendar you can see no matter where you are in the world).
Keeping them around is a sensible fail-safe in case some nuclear power sets us back into the dark ages.
I’m pretty sure that “oh, shoot, things got wonky… toss a 13th month in here real quick” is due to people trying to force months to fit weeks.
It’s the opposite of what I was saying about the role that months play in timekeeping & how they work.
ALSO, the same can be said for weeks & leap days… so if it’s a point against months, it’s just as much a point against weeks.