The problem is that the church calendar, at least in the Byzantine Rite, doesn’t work if you use a different calendar for the fixed feasts (like the Nativity, December 25) and for calculating the date of Pascha.
So if you use the Julian calendar (13 days later from the Gregorian calendar right now) for figuring out when Pascha is, then you should also be using it for calculating “what day it is” the rest of the year.
And THAT means that today is November 15,even though the banner on this website says it is November 28. And while I understand the (somewhat self-serving) appeal to this - we get good prices on Christmas ornaments! We can eat meat on American Thanksgiving! etc - it introduces a major mental distinction between “my life in church” and “my life as a whole” which I think is somewhat unhealthy.
“Well, when we’re in church, today is November 15. But when we go outside it is November 28. Except at meal times, when you have to remember the “true” day so you know whether to fast, and what saint’s day it is. But if you are at the after-Liturgy luncheon and Father announces the date of the next church social, he is probably using the “non-church” date for that.”
And so on. Not to mention that God has arranged it so that the first day of Spring (astronomically) now falls on a day other than the one used to calculate the Orthodox Easter - so even the physical world falls into the “no working according to God’s design” category. I have heard some Orthodox use the Julian / Gregorian calendar to help children remember how completely screwed up everything outside “the church” is. Personally, I think Christ (with our help) is redeeming the world - but does that mean we are called to see that civil society adopts the Julian calendar, so that they finally see that today IS November 15??
P.S. I have NO problem with the churches in one region deciding to use the Julian calendar in common. But simply adopting it for the entire Christian world raises the very real issues I have outlined.