Yes, you're missing something obvious. The satellites aren't all located
due south of your home. They're all along the celestial equator (more or
less, allowing for parallax, which is why equinox blackouts don't happen
exactly at the equinoxes), but the celestial equator passes through the east
and west points on your horizon as well as the point due south whose
altitude equals the complement of your latitude. So you could have a dish
pointing due east or due west, though the signal strength would be pretty
poor. These days most commercial TV broadcasts occupy the same one or two
satellites, but in fact there are probably satellites at most of the allowed
orbital positions. (Only a few degrees seperation are necessary for a large
dish. High-power satellites must be further apart since the small dish
doesn't focus as well.)
--"Volleyball is racquetless team ping-pong played with an inflated balland an elevated net while standing on the table." --George Carlin