Why would the sunrise align with the same point at the same time each year?

It doesn't, quite. But the difference in the direction of sunrise for the same date each year is so small that you can't really measure it.

As to why the direction remains the same each year, it's because a commission appointed by Pope Gregory in 1582 set up the Gregorian calendar that way. The older Julian calendar (devised, legend says, by Julius Caesar) did not properly adjust the calendar for leap years, and by the year 1570, the conventional calendar in use by the Church was out of synchronization with the seasons. The Gregorian calendar properly adds leap days periodically to the calendar to keep the calendar aligned with the seasons.

So, the same day of the year will correspond, within 3/4 of a day, to the position of the Earth in its orbit around the Sun.