Even without that issue, many Orthodox Christians still use the Julian rather than the modern Gregorian calendar for religious purposes (while still using the Gregorian for daily life). That's why Orthodox Christmas and Easter are on different days than the Catholic/Protestant ones.
Even the Roman Catholic Church that uses the Gregorian Calendar has moveable dates: Easter Sunday always falls on a Sunday and not on fixed day of a fixed month. On the other hand, Christmas day is always on December 25. Go figure.
You would end up with the "religious week" and the "standard week" moving relative to one another over the years, in a classic case of trying to fix messy details and thereby adding more details (as in https://xkcd.com/927/).