I would welcome rail from here to Nashville, I don't get up there much primarily due to traffic. I've lived in many places, including Manhattan and San Francisco, and I sorely miss the convenience of the train.
Simply put, rapid transit would be great for commerce in Nashville and it would either alleviate in some way or at least mitigate traffic congestion that will be compounded by the rapid growth of Nashville's suburban communities. I think costs of extensions into these communities should be shared, but ultimately funding rapid transit through sales tax makes sense because commerce/tax base should increase.
Anyway - just came here to give some perspective - don't get the impression that Williamson County, Tennessee is full of racists. It is not as diverse as Davidson County, but nonetheless a wonderful place to live.
EDIT: I forgot to mention that I'm anxious to see how autonomous cars will affect situations like the one facing Nashville.
This alone might make it appealing to take normal methods (car or autonomous car) into the city for leisure.
Ah but what will that effect be? There's nothing about a self-driving car that increases road capacities. And since a self-driving car doesn't get bored or frustrated, there's less of a disincentive for a self-driving car to drive during rush hour. So it seems like if all else remains equal, self-driving cars are going to make traffic worse, not better (sure, if you could combine them with ride-sharing, they might reduce things but lyft and Uber right now could be ride-sharing but they aren't. No reason to expect a self-driving taxi would better unless something impels it).
Now, if these self driving cars were high passenger capacity, that might have a great effect- but of course then we'd just call them a bus.
As for the support from outside Nashville proper, I think that all of my friends and coworkers who live outside the city have much more to gain from a transit system that extends outside the county. My commute is pretty reasonable, a consistent 15-20 minutes. My coworkers in Williamson, Rutherford, or Cheatham Counties all have significantly longer commutes. I'm not saying support is a given, but for all the people who talk about Nashville turning into Atlanta, it's not those of us who live inside Davidson County that have the most to lose.
(I also live in Davidson county and even ran for school board).