Why does the Gregorian calendar start where it does?

January 1^st doesn't seem to have any sort of significance with any celestial phenomena.

Not in general, but the first January 1^st of the Julian calendar (i.e., January 1^st, 45 BC) was indeed synchronized with the first new moon following the previous year's winter solstice :

It was probably the original intention of Caesar to commence the year with the shortest day. The winter solstice at Rome, in the year 46 B.C., occurred on the 24^th of December of the Julian calendar. His motive for delaying the commencement for seven days longer, instead of taking the following day, was probably the desire to gratify the superstition of the Romans, by causing the first year of the reformed calendar to fall on the day of the new moon. Accordingly, it is found that the mean new moon occurred at Rome on the 1^st of January, 45 B.C., at 6h. 16′ P. M. In this way alone can be explained the phrase used by Macrobius: Annum civilem Caesar, habitis ad lunam dimensionibus constitutum, edicto palam proposito publicavit. This edict is also mentioned by Plutarch where he gives the anecdote of Cicero, who, on being told by some one that the constellation Lyra would rise the next morning, observed, Yes, no doubt, in obedience to the edict.

— William Smith, A Dictionary of Greek and Roman Antiquities, page 231, John Murray, London, 1875.

It is not that simple. According to this page,

http://www.webexhibits.org/calendars/year-history.html

it has nothing to do with Gregorian calendar and was actually decided by Iulius Caesar. Later in various countries we see many various dates when the year starts. March 1 was also a very common date and was used in ancient Rome. Starting a year from some astronomical event (like an equinox) is inconvenient because the true year contains non-integer number of days, so the date of the equinox is always sligthly floating.

The first time the new year was celebrated on January 1st was in Rome in 153 B.C. (In fact, the month of January did not even exist until around 700 B.C., when the second king of Rome, Numa Pontilius, added the months of January and February.) The new year was moved from March to January because that was the beginning of the civil year, the month that the two newly elected Roman consuls—the highest officials in the Roman republic—began their one-year tenure. But this new year date was not always strictly and widely observed, and the new year was still sometimes celebrated on March 1.

http://www.infoplease.com/spot/newyearhistory.html

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.