I had an hour-long debate with coworkers about the International Fixed Calendar (Sol Calendar). There are so many benefits! I particularly like that every 1st of every month is a Sunday, every 2nd of every month is a Tuesday, etc. No more "What day or date does _____ holiday fall on _this_ year?"
What you're saying is apparently true for the International Fixed Calendar, but not the Sol Calendar. According to [1], the first (and second and third...) of every month in any given year is the same day of the week, but the mapping changes every year.
The International Fixed Calendar may start every 1st on a Sunday but the European Fixed Calendar will probably make each 1st a Monday. Agreeing on things is hard.