The Roman calendar (on which the Gregorian calendar was based) was synchronized with a celestial event: the lunar cycle.
Originally, each Roman month began on the first day of the new moon. However, it was timed according to the full moon, which fell on the ides. The reason for this is that the full moon occurs on only a single day, but a new moon can last several days. Therefore, originally the key event was the full moon, because it was unambiguous. Between the new moon and a full moon are a little over 14 days. Therefore, originally the Romans made the Ides fall on the full moon and the Kalends (the first day of the month) fall either 13 days or 15 days before in a schedule designed to average out correctly.
In the reign of Numa Pompilius the calendar was regularized so that this synchronism was no longer kept.
The Roman ritual New Year occurred on the Kalends of March because March is the month in which the Spring solstice occurs. The civil year was switched to begin on the Kalends of January in 153 BC for political reasons: the Senate wanted the new consul Quintus Fulvius Nobilior to attack the city of Segeda during the Celtiberian Wars immediately. Therefore, they moved up the day of the appointment from the Kalends of March to the Kalends of January.
So, to summarize, the short answer to your question is that the months were originally aligned to the day of the full moon, but when the calendar was fixed to the solar year, this synchronization was lost. The reason why the solar calendar was not aligned with the solar event (solstice) was that Numa Pompilius kept the sequence of days roughly as they had been under the lunar calendar. If he had changed the Kalends of March to fall on the solstice, then things might be synchronized in the way you suggest, but he probably did not want to disrupt the flow of days as it then existed.