The central region of the galaxy is in the direction of the constellation Sagittarius and not far from Scorpius. It is near them that the plane of the galaxy intersects the ecliptic (plane of Earth’s orbit) at an angle of about 60°. If the planes were nearly parallel, then we would see the Milky Way running through all of the constellations of the zodiac.
Picture a summer evening while the region between Sagittarius and Scorpius is crossing your southern meridian. The ecliptic would appear parallel to the horizon. You would observe the Milky Way sweeping upward at an angle of about 60° to the ecliptic.
You mention the Earth’s hemispheres, but they are separated by the Earth’s equatorial plane, not the ecliptic or galactic planes. The equatorial plane currently intersects the ecliptic plane at more than 23°. In the current era, the center of the galaxy is visible from the Earth’s South Pole and not the North Pole. That will eventually reverse due to precession. But in either case, observers on most of the Earth’s surface experience periods of the day during which the galactic center is above the horizon.