I approve of this with one change: the festival would last to the second exactly as long as it would take to synchronize the calendar to our orbit around the sun (no leap years or anything).

Could you imagine a little more than five days with no representation on the calendar? Truly a "let everything go" period of time. Society would fall into chaos, no appointments could be made for those days because there is no way to address an exact point in time before the calendar reestablishment; truly an environment for a giant worldwide party.

In reality tough we would just call it the 13th month, or the minimonth and everything would go business as usual :/

This is the Alexandrian calendar, proposed in 238 BC and adopted in 25 BC. (Previous Egyptian calendars had a fixed 5 day intercalary festival.) It is still used by the Coptic church and parts of Egypt.

Not sure if you know this, but this is exactly how the Republican Calendar worked during the French Revolution. The bonus days were known as the sans-culottides (meaning days of the "sans-culottes," the self-chosen name for the Revolutionaries reinforcing the difference between them and the nobility) and celbrated Republican values like Labor and Virtue.