I'm skeptical of this. I've not lived in Nashville, but I've lived in a whole lot of cities across the South: Charlotte, Atlanta, New Orleans. What almost all Southern cities have in common is that conservatives are a minority within the city limits. And indeed, when I check the 2016 election results for Nashville, I see that Clinton won against Trump 59 to 34% -- almost double! And also not surprisingly, Nashville has a Democratic mayor.
Does that sound like a city with right-wing politics forcing itself on every space?
The fact is, unlike Southern small towns and rural areas, Southern cities are in general fairly moderate to liberal, with a significant conservative minority (say 25-30%). Now, given our current political situation, that minority may be at present more vocal than usual, but that doesn't mean that the city is beset by "right-wing politics" occupying every sphere of life.
Also, I've lived in working class neighborhoods in Southern cities for most of my life and have never heard anyone blasting evangelical sermons, around my house or around friends. That's 20+ years as an adult. So you picked an extraordinarily unlucky Airbnb.
Yes, you run into occasional racists or homophobic assholes, but I occasionally ran into the same kinds of people when I lived in NJ for 2 years. There are assholes everywhere.
But by far most of my co-workers working in universities and small companies have been either moderate or liberal, with an occasional conservative thrown in. Which isn't surprising, since that reflects the city's political breakdown. But (at least until recently) we all tend to get along.
Now wrt to the running, I have no idea, you may be right there (although I know that the Chattanooga area has some great running).
But wrt to politics, I'm fairly convinced that your friend's experience in Nashville is either a misinterpretation or they've met with an outlier company and friends (eg, did they work for a defense-related company? if so, those are conservative everywhere, and has nothing to do with Nashville).