You are not the only person who is frustrated that the gospel of Luke does not say:
On the fourth of May in the Roman year 773 AUC , the word of God came to John son of Zechariah in the desert.
—Luke 3:1-2, Imaginary Text
But there is a very good reason why the New Testament does not give precise dates—there was no single calendar in universal use. Yes, it is true that Julius Caesar had standardized the Roman civil calendar in 46 BC, but it was a lot like the metric system in the United States today: official, but largely unused by ordinary people. Instead, everyone used their own local calendar.
Let’s image a traveler in what we would call the year 20. In Egypt, the year begins on the first of Thoth, toward the end of summer. Our traveler sets sail from Egypt on that day and arrives in Greece. On his arrival, he discovers that the locals consider it to be the middle of the month of Boedromion, the third month of the year 332. Because he is an Egyptian, he’s accustomed to the civil day beginning at midnight, but in Greece, they don’t advance the date until sunrise, so he has to take that into account. After taking in the scenery for a day or two, our intrepid traveler sets sail for home, making a stopover in Joppa. Here he finds that the locals consider it to be the month of Marchesvan in the year 3831. He also finds out that the date changes at sunset, not at sunrise as in Greece, or at midnight as in Egypt. So he asks a Roman government official about the correct date. The official replies that it is September in the year 770.
This is not an unlikely scenario, not just because the Mediterranean Sea is small and can be crossed rather quickly by sail, but also because there was tourism in ancient times. Modern archaeologists can attest that Roman tourists of the first century left graffiti on ancient Egyptian monuments. Modern tourists have to adjust their watches to the local time zone, but ancient tourists had to adjust to completely different calendars.
Some years ago, someone made a movie about those whirlwind tours of Europe. The movie was entitled, If It’s Tuesday, This Must Be Belgium. If that movie had been made 2,000 years ago, the title could have been, If Today’s the Twelfth of Elaphebolion, This Must Be Greece.
So to avoid confusion, ancient documents intended for wide distribution often dated events by referring to the names of officials who were in office rather than by using a calendar date. The New Testament reflects this practice.
In the fifteenth year of the reign of Tiberius Caesar—when Pontius Pilate was governor of Judea, Herod tetrarch of Galilee, his brother Philip tetrarch of Iturea and Traconitis, and Lysanias tetrarch of Abilene—during the high priesthood of Annas and Caiaphas, the word of God came to John son of Zechariah in the desert.
—Luke 3:1-2, NIV
You can read about the history of our calendar.