How come official recording of time and years started off with the birth of Jesus?

Originally, years weren't tracked at all. Later, local chronology was calculated by the years in the reign of the local ruler, so that the same year could have different points of reference throughout the world. It wasn't until the Venerable Bede, a monk in Jarrow, England attempted to discover how long before Christ had been born that the concept of Anno Domini (AD, the Year of our Lord) and Before Christ (BC) came about. He was off by a small margin of error, but we still use the basic schedule resulting from his calculations.

BTW, there was no Year Zero. Using Bede's basic timeline, we go directly from 1 BC to 1 AD. Yes, 1 AD was the first year of the first century AD. The 25 December was just a handy assimilation of pagan solstice ritual given new meaning; no one really knows exactly what time of year Christ was born.

Well, you're assuming that everything the Romans recorded survived. Obviously the opposite is true, most of the official documents were lost to time. Even if there was an official birth/death/court record of Jesus somewhere, there's no reason to think it would have survived. A better argument would be to point out the absence of Jesus from mid first century Jewish historians like Philo and Thallus.

cuz jesus is awsome !