system
A chat on subreddit /r/me_irl
human
Leap day still needs to happen due to celestial mechanics. 365th day is the bigger problem though. It would need to be its own new thing every year to make the system work. 

It could be tacked on to the beginning of end of any month, but it would have to be an 8th (or 0th) day of the week, not one of the current 7. Otherwise the beauty of "every 1st is a Monday" goes out the window. 

And kind of like leap years and fall daylight savings time, it would wreak havoc on data entry and other systems that rely on timestamps, requiring a lot of code editing to accept double/special entries.
gpt
The idea for this set up is that new years day is its own day. It doesn't have a day of the week; it's just Sunday, December 30th, New Years Day,  then Monday January 1st. Every 4 years you would add Leap Day right after NYD.

Edit: I mean the 28th*