I just pointed this out to a friend and she gladly brought up the Shire calendar from Tolkien's works, which according to her has 12 months, 30 days a month, 7 days a week, 365 days a year with the date of each month staying the same weekday due to the separation of the five other days as holidays that are separate from the months (including leap-day, which doesn't belong to any day of the week).
Which actually solves the exact same problem in a much cleaner fashion than this proposal.
No, it would not be accepted. The 7 day cycle is strict - you can not add or remove days from it. It might be fine for work, but it will not be fine for religious observance.
If you tried then those who observe religious days will keep their cycle, and will not go to work on the real weekend (even those who otherwise wouldn't care would make a point about it), which will be out of sync with the new calendar weekend. It won't take many people doing that for the calendar to fail to be accepted.
The real innovation in this calendar, which I don't rememebr seeing in previous perpetual calendar proposals, is the system for 7-day-week seasonal drift adjustment. Adding an "extra" 5 days to every year is much more common, but not popular.
What I can't figure out is what algorithm they use to calculate when to add an extra week. It seems slightly irregular, but I asssume there must be a simple formula I could use in a date library to account for it.
Well, with 364 days that means there's 1.25 days pushed into the buffer every year. That means that the buffer would be full (mathematically) every 5.6 years hence the "every five or six years." In order to keep general season alignment, I figure they vary between the two. This decimal doesn't go away until after 28 years - so I assume some combination of fives and sixes that add up to 28. Lets say three sixes and two fives. (18+10). So in order to make any pattern they might do something like 6 5 6 5 6 6 5 6 5 6. And now, looking at the leap years, I think my theory is confirmed: 2015-2020 (5)