The article fails to make the distinction between lunar and solar calender. It's just about dividing the solar year to equal months, but we also want to make the month align with the moon, which circles earth for 29.5 days on average. The big problem with the Gregorian calendar, in my opinion is that it so clearly missed the lunar month. it just feels intolerably close, but not quite there...
by the way, no need to invent a new calendar. the Hebrew calendar is both solar and lunar.
I suspect the authors intention here was to present a simplified calendar with some nice guarantees (e.g. dates matching days), not a lunisolar alternative. The Positivist calendar ignores the lunar cycle to reduce complexity that others have in spades, including both the Gregorian and the Hebrew calendars.
It's actually unclear to me why we particularly care about how long it takes to orbit the Sun. Axial rotation, sure, the length of a day is important. But outside of that... why do we need particular units like weeks, months, and years at all?
Religion and seasonality. 7-days as a unit of measurement is written into unamendable documents hold as holy by all Judaist monotheisms (Jewish, Christian and Islamic). The ciclical experience of seasonal-weather and daylight-lenght changes are also too powerful to deny.
AFAIK, Islam traces its roots back to Abraham, who was Judah's great-grandfather. So it would seem more appropriate to call it an Abrahamic religion, no?
by the way, no need to invent a new calendar. the Hebrew calendar is both solar and lunar.