Except that while it's very hard to get it exactly right, it's really easy to get close.

Getting to 360ish days is relatively trivial - as best as we can tell, a 12-month, 360 (+- 10) day year was developed independently in at least India, China, Mesopotamia, and Mesoamerica - and that's a minimum: there's reason to believe that northern Europe, Mediterranean Europe and/or North Africa, Southeast Asia/Polynesia, South America, North America, and southern Africa also independently developed calendars that were not far off from 360 days. From a 360-day calendar, all you have to do is throw in a few non-year Holy Days before New Year (which, for many calendars, happens at one of the four main days - the solstices and equinoxes) until it's the right time, and move on. Alternatively, if your calendar is lunisolar, you add a month if New Year is too early.

Yes, getting things exact is really hard - even the Julian Calendar is off by a little bit, and technically the Gregorian Calendar is off by a little bit (it's off by .0003 days per year - or about one day in 3000 years). But I think you're selling human intelligence short saying it's not trivial to get a working calendar - as long as you understand the calendar isn't perfect, being a few days off isn't a problem. And both ancient calendars still in use (Chinese, Hebrew) understood that, and had rules to make sure that the calendar was not off by more than one month - and it's safe to assume they weren't the only ones like that.