I had a bit of a left-field thought today. Have they intentionally screwed with the calendars to promote the idea of climate change, global politics, and to disconnect people from the sky, stars, and their natural environment.
I propose that the length of the true year is between 360-364 days in length. I think 364 is the most likely.
364 / 28 = 13
There are 28 days to the moons cycle, and this would break nicely into 13 lunar months. I think we are supposed to use the moon to tell dates and time alongside the stars, not the sun.
The sun is being used to throw everything off. It's part of their solar worshipping. They changed the calendar to be solar and adjusted to make it fit, but the sun was never supposed to be used for time keeping.
It is causing an extra 1.25 days every year to shift the actual seasons around. Very subtle, and not really enough to notice except over a lifetime - which would only be corroborated with at least 60+ years to truly notice as your idea of the months would be respective of your birth year.
This was likely done around the 1700's and would have coincided likely with the American/french revolution (revolve) and it pushed the ideas to the forefront of society.
Since this would result in approximately 100 years seasonal shifting, you could setup narratives for climate change (cooling and heating) because they are being told that December is supposed to be a certain way when in fact it's supposed to be tracked with 13 lunar months instead.
Ultimately I think the year would have to be based off the sun because seasons are based off of when the sun's declination reaches the equator, or tropic of cancer, tropic of Capricorn. Anyone can use a sextant to check where the sun is on its annual route over earth, or monitor hours of daylight in their location. FYI I am a flat earther, but it's the sun's declination over earth that dictates seasons on both models, so it doesn't really matter. That being said, the sun might dictate seasons/years, while the moon simultaneously dictates months. I recall the ancient Hebrews had a year that was 364 days as you said, with one or more neutral days in between to ensure the seasons don't become misaligned with the years. I don't know if anyone is sure how these neutral days are calculated. They have certainly screwed up our calendar with a 365 day year, and leap years to continually throw us off.
I had considered adding a 'new year day' but the months need to coincide with the moon, so I feel like you cannot inject a day unless you do so in the length of your day as well. Ie....you don't use our clocks either to tell the time.