I'd suggest reading this article (link)
for some context on blacks on television in the 1960s.
From reading it, I suspect Star Trek
was never in serious dangerous of not being carried in "the south" because of a continuing character like Uhura. After all, as the page indicates...
When I Spy premiered, NBC officials seemed pleased that only three stations—in Savannah and Albany, Georgia, and Daytona Beach, Florida—refused to carry the show. It was seen, however, on 180 other stations covering 96 percent of the country.
...and I Spy
had a black co-star.