As someone who has proposed this system myself, I feel the need to point out that the meme is glossing over a couple key points:
First and foremost, 13*28 is 364 days, so to avoid slippage you’d need an extra day appended to every year, either as part of a month, which breaks symmetry, or on it’s own. You’d also still need leap years.
And in order for the days of the week to be immutably aligned with dates, these extra days would also have to not be part of any week. Which is a big problem if you want to get anyone who practices an Abrahamic faith on board with the plan.