I understand this is a bit of fun trying to rethink a very old thing, but the benefits are really bare even when looking past all the damage of a potential transition.
Basically, you're consolidating the unevenness between years/months/days of the week to one special 1 or 2 day period. Ok, that's alright, but it still requires a bit of extra help. You still have to think through calendars understanding "Oh! I have to add a day for the epagomenal. Oh, it's a leap year, so it will be 2 actually."
That might be preferable to having to think through 31 or 30 or 28 or 29 days, but not by that much. Is aligning months and days of the week to the same time every year a good thing? I suppose it's easier to determine what day Thanksgiving is every year, but otherwise, it seems like a minute benefit. On the other hand, I wonder if there are some benefits of that not happening. The only one that comes to mind is it would suck for your Birthday to be on a Monday every year, but I suspect there might be other reasons where enforcing a day to occur on the same place in the week could be harmful.