I mean, for the most part. You would have 13 months of 28 days with 1 leftover. Make that one new year or something. Leap years would just have another straggler day, lump that in with new year I guess.
The moon thing is wrong though. The moon does not operate on a 28 day cycle.
As for Monday being the 1st and Sunday the 28th, that wouldn’t matter at all. Any day could be the start of the month.
If you buy something for a cat, they won’t use it.