I've known about this for years. I think Arthur C Clarke used to write about it in science fiction books I read as a kid.
I've promoted the idea to people occassionally, the way I look at it is a double sunday at the end of the year. Side note: The weeks should start on a Monday. I think of Sunday as the last day of week.
It is more ordered in the sense of fitting weeks and months into years and weeks into months. Months of 28 days works well.
There is a problem with it which is that 12 months can be divided neatly into halves, thirds and quarters (including company's fiscal quarters and seasons) while retaining whole months.
So with a 12-month quarter a company's fiscal Q1 can be Jan 1st-March 31st - nice and neat. With this system quarters would be a mess. Similarly, you wouldn't have the ability to have summer as June, July and August and winter as December, January, February etc (although a lot of people insist that the season start on the 23rd or 21st of the month, but I don't like it).
Changing this system would cost a huge amount of time, money and hassle. People's birthdays would have to be changed, and everything scheduled for a future date, from eclipse predictions to elections and a thousand other things would have to be redone, it would be a total mess. Some people get confused just when the clock changes by one hour!
So because it has some advantages, but some disadvantages, and it would be a tremendous hassle to change, it's probably better to stick with the current system.
I do think if the system you proposed already existed, and then someone proposed changing to our actual current system of random months of 28, 30, or 31 days that would be even more of a non-starter. So the 13-month system is probably slightly better, but not enough to be worth the hassle of change.