There are dozens of calendars around the world that have nothing to do with the birth of Jesus Christ.

In India there are at least a dozen Hindu calendars. The two most commonly used are the Vikrami calendar (in north India), which begins in 56 BC during the reign of King Vikramaditya, and the Saka calendar (in south India), which begins on the vernal equinox in the year 78 AD. These are both modified lunisolar calendars, based on lunar cycles but corrected for solar cycles. They are not merely of historical interest, they are in widespread use on a daily basis in modern India. Many Hindu festivals are based on calculations off these calendars, and they are regularly published in the millions and bought by people to hang on their walls.

In Islam there is the Hijri calendar, which is based on the Hijra - Mohammad's journey from Mecca to Medina in 622 AD, which is its starting point. This is a lunar calendar, and as such it varies from the solar calendar by increasing amounts every year, until it eventually cycles back to the beginning. It is not really an astronomical lunar calendar, but rather a legalistic lunar calendar. While an astronomical calendar would simply follow the phases of the moon, the Hijri calendar adds the qualification that the moon must be observed by "trustworthy men". So while a month typically begins at sunset on the observation of the crescent moon, if there are clouds or if no trustworthy men are around, it may be delayed to the next sunset. So months can have 29 or 30 days, but it's impossible to predict in advance how many days a given month will have. For this reason, the Hijiri calendar of different countries might fall out of synch, with Eid being celebrated on one day in Saudi Arabia, for example, and on the next day in Morocco. Nevertheless, the Hijri calendar is central to Islam and is widely used across the Muslim world.

China also has dozens of different calendars, but the most commonly used local calendar is the Han calendar. Like the Indian calendars, it is also lunisolar, and is used for traditional stuff like figuring out auspicious days for major events like weddings. The Han calendar doesn't really have any start date, it has been modified, adopted and readopted over time and there is no clear continuous use from any defined starting point. In the early 19th century, it was arbitrarily decided to begin it from the birth of the Yellow Emperor. This is a semi-mythical figure who supposedly began Chinese civilization, and his birth is given as 2491 BC for the calendar.

There are dozens of local calendars in use across southeast Asia. Most derive from some version of the Hindu calendar, though a couple are of Chinese origin as well. They are generally lunisolar.

Why, for example, do the Chinese now use this system if their culture is more based around Buddhism?

Pretty much all countries that use local calendars also use the western calendar, and the western calendar is considered the official calendar for government purposes. The reason, of course, is standardization. It would cause an awful lot of confusion if the world couldn't agree on some common calendar, there would be endless translation of dates between different systems with the possibility of mistakes every time. Since the Europeans colonized much of the world and spread their Gregorian calendar to all these places, it stands to reason that this is the one calendar that has been introduced to the most different countries, so it's easy to standardize to because of past familiarity with it.

I don't know if it will have exactly what you are looking for, but I would recommend checking in the FAQ. There are several questions related to this topic. In particular, I think this thread may be of interest.