Some people, groups, or governments could change over on a case-by-case basis to a week-based system, without a sudden change to the worldwide date system. Just label the weeks with numbers, e.g. 1-Wednesday, 27-Monday, or 52-Sunday for Christmas this year.

It seems Chinese people generally don't remember the time of week so much as the Month and date. Whenever someone in China tells me "this is happening on the 28 November", in my mind I have to work out what day of week it is before I can internalize it in my memory. Chinese seem to just remember the month and date. Schools often say "Monday's and Tuesday's timetable will be shifted to the previous Saturday and Sunday, respectively".

And of course their idea is Western-centric. Chinese and Muslims both have their own calendars for festivals.