Didn't the French try to change the year to 10 months during the French Revolution?

Regardless, 12 is a better number of months. You can easily divide it into halves, thirds, and quarters for nice groupings of months. 13 you can't divide by anything.

Exactly. The Gregorian Calendar is an abomination. We should have 13 months and follow a true lunar calendar.

You still get a remainder of 1.25 days each year