Remember that most of those sodas were originally meant to be medicinal and were created by pharmacists (Dr. Pepper legally isn't a cola, but it was still created by a pharmacist). Most of the other old non-colas were created by pharmacists, also, and were created before Coca Cola: Vernor's (1866, Detroit), Hires Root Beer (1875, Philadelphia) and Moxie (1876, Massachusetts, actually a doctor, but you get the idea).
But except for Dr. Pepper, the South didn't produce many early non-colas. Big Red didn't show up until the mid-30s, 7up (in St Louis) around 1929, and Ski until the 50s.
So, I guess, why did the South focus on Cola at the expense of other flavors, and why did the rest of the country focus on other flavors? I don't really know. My guess is that it's sort of like New York Pizza and the Chicago Style Hot Dog. One person came up with the idea, it became popular regionally, and everyone else jumped on the bandwagon.
Some pharmacist in the south came up with a cola syrup and it was a hit. Regional travelers had a taste and found it cured what ailed them, went back home and asked their own local pharmacist to come up with something similar. Coca Cola was, to a certain extent, created in this manner: John Pemberton created a knockoff of the very popular Vin Mariani (wine+coca leaves) by adding kola nut extract. The county he was in went dry, so he took out the wine, tweaked it a bit, and lo, we have Coca Cola. My guess is Pepsi and RC came about because of the popularity of Coca Cola, sort of like why we have Giordano's and Edwardo's and Lou Malnati's, in addition to Pizzeria Uno.
This is all speculation on my part, but it makes sense..