Why doesn't the sun reach its highest point overhead at exactly 12:00 noon?
It's not a question of latitude but longitude and time zones.

Prior to the advent of railway transportation the time set on clocks varied from towns within close proximity because the clocks were set to noon when the Sun was at its highest in the sky. This was more apparent for towns east and west of each other (ie different longitudes).

When railway transportation was developed there was a need to standardize times so that train timetables could be developed and arrival and departure times would be accurate and consistent

This precipitated the development of time zones. Each time zone is approximately 15 degrees of longitude wide, but there are differences because of national/regional priorities. China which has a similar longitudinal extent as the contiguous states of the US has only one time zone, whereas there are four in the contiguous states of US.

Within a band of longitude the time is the same. In the time zones immediately east and west, the time is one hour more (east) and one hour less (west) - ignoring difference due to daily savings.

The reason for time zone being approximately 15 degrees of longitude wide is the rate of spin of the Earth is 15 degrees/hour (360 degrees every 24 hours).

You state that in your town, noon is either 12:30 or 13:30, depending on daylight saving time. The reason for noon occurring at half past the hour is due to the boundaries of the time zone where you live. Where I live, noon is approximately at quarter past the hour.