Sirius is close to us so its actual position is going to be very close to its apparent position. Why do you think that it would move into the Southern Hemisphere?
The Sun is even closer, but you say the Sun's apparent position is off by 2.1 degrees from its actual position. Do you believe the Sun revolves around the Earth?
No, I'm not crazy. But you say that maybe half the stars mrjesse sees when he looks up at the night sky are actually on the other side of the world. How would they get there without travelling faster than light?