I think a 13-month calendar is a non-starter. A 12-month year can be divided evenly into halves, thirds, and quarters...a 13-month year cannot be divided into any smaller units without splitting months.
National Semiconductor used a 13 month calendar way back when I worked for them.
They didn't have months in their corporate calendar, they had 28 day 'Periods' 3 Periods to a quarter, with the extra Period stuffed into quarter 4, where it helped increase the year's bottom line, and offset lost productivity for Christmas shutdown.
It was a great system, for all I know they are still using it.
**Yes, and if each month begins on Sunday we could have Friday the 13th 13 times every year.**
LOL!
The prime reason.