Not a calendarologist, but I'm pretty sure lunar calendars were tried and rejected for a reason. Other than the places that still use them for traditional reasons.
Of course, maybe they just didn't have the concept of leap days?
It only comes out to 364 days so you'll still need to handle that 1.25 extra days in a year otherwise it'll drift. You could just add December 29th as a special day past Saturday, but then you lose sync with the moon, eg.if New moon was on Sunday the first in the previous year, New moon would be on December 29th instead of on Jan 1st the next year and all new moons would be on the 28th.
You can keep your calendar in sync with the moon or the sun but not both.
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.
What exactly would be so troublesome with having a "special day" outside of the usual week/month cycle? You can still go worship on whatever day of the week applicable to your faith. Just make the last day of the year its own thing. We can call it "New Years Eve" and party together.
It makes the calendar less than compatible with the commandment to keep the Sabbath by not working on every seventh day.
Which is not insurmountable in practice (e.g. by keeping a separate ecumenical calendar) but you can bet it would be a significant source of opposition.
But you'd be resting on the extra day, too (if you want, I for one will be partying). So you'd never go more than seven days without rest.
If that still doesn't fly, I suggest we combine the extra day and the previous day into a mega-day that is 48 hrs long. Then everyone except programmers will be happy (we're never happy with datetime conventions anyway, so what's one more if-else statement between friends).