Even the Gregorian calendar is anachronistic.

"An anachronism—from the Greek ανά (ana: against, anti-) and χρόνος (chronos: time)—is an error in chronology, especially a chronological misplacing of persons, events, objects, or customs in regard to each other. The item is often an object, but may be a verbal expression, a technology, a philosophical idea, a musical style, a material, a custom, or anything else so closely associated with a particular period in time that it would be incorrect to place it outside its proper domain."

A year to year start/stop 'should' start on Dec. 22.

Consider that the winter solstice, the shortest day of the year, is on Dec 21st. The next day is the date that the functional days start to grow longer.

Oh, and the Roman calendar was short by 5 days, they knew it but just declared it the national holiday Saturnalia. 5 days of eat drink and be merry. One of the sub holidays within was Juvinalia(sic?). the only Roman holiday for children.