The main hiccup is the system is off by a day. Some people “fix” this by saying the extra day should be “new years day” or something similar that exists outside the main calendar and doesn’t have an actual date or day assigned to it. Personally I think that’s kind of silly but it does work.
The second problem which to me is a much bigger problem, is he argues every month starting on Monday is a feature, I think it’s a bug. The result of this is every date is the same day, every year. If you are born on a Wednesday, your birthday will always be on a Wednesday. I like it mixing up and getting to have your birthday on different days.
Also almost everyone will have a new birthday they have to learn and too many people would simply be unwilling to go along with that.
And all that is ignoring the monumental task of changing every computer system in the world.
Edit: also 13 is just kind of a rubbish number to work with and doesn’t divide into anything nicely.