I've often dreamed of and revisited this idea. I first started thinking of it seriously when I realized I was paying the same rent in February as in January despite a significantly shorter-than-mean (30.4375) month...!

My ideal year is 12 months, each 5 weeks long, each week 6 days long. At the summer solstice, 3 intercalary days (bank holidays), at the winter solstice, 2 or 3 intercalary days depending on leap year.

I thought it was 13 months exactly 4 weeks long, which takes us to exactly 364 days

There would be 5-6 days without time every years?

Yes, see also

https://en.wikipedia.org/wiki/French_Republican_calendar