It wouldn’t be perfect, but you would always know which day of the week the date was.
For example, Sundays would always fall on the first, eighth, fifteenth and twenty second of each month.
Thirteen (months) times twenty eight (days) equals 364. The extra day needed to round out the year could be celebrated as a holiday between day 364 and day 1 of the next year.
Our current calendar was introduced in 1582, over 400 years ago. I think we could do better.
What would you call the 13th month?