Currently, everyone in the world agrees about the days of the week (correct me if I'm wrong). If it's Monday in France it's Monday in Finland, besides a few hours due to timezones. But if a particular society adopts this system you describe, or any system under which every year starts on a particular day of the week and is solar aligned, that necessitates having an incomplete week and losing that sync with the entire rest of the world.

A possible solution is to only use leap weeks. So every year has 364 days, but every 6 years or so (spare me the exact calculation) you track on a leap week to realign with the solar cycle. This is similar to the leap month in the Hebrew calendar - months follow the moon so a leap month is the smallest unit possible to tweak the length of a year.