They don't seem to have these problems in Murfreesboro or Williamson County even with BA. It comes down to local governments period. How bad do they WANT to have great school systems?
Also Ensworth, MBA and BA amongst others, only have a very small percentage of Metro Nashville kids. Think about it, Ensworth probably has 20 kids, MBA probably 10 or so in their entire system and BA maybe 30 tops. That's it and we're talking 7th-12th grades. Privates have very little effect on metro schools IMO.
Furthermore, most inter-city kids (or middle class kids, which most of the kids in question in truth are) can't afford to attend MBA from PreK-12. I've never heard anyone say "Richard Bowers should have QB'd Hillsboro High to two state championships instead of Ensworth" or "Conner McDermott could have been the best TE at Hillsboro High". The issue is only raised in reverse questioning the merits of inner-city kids and parents. The "other" kids at these schools are also assigned to public schools.
Food for thought: the best football player (Corn Elder) and the best basketball players (Elder, Jalen Lindsey) of recent years, played for schools inside of Metro Nashville. But, they are both are from Murfreesboro. Did that destroy the fabric of football and basketball in Murfreesboro? No. Not by a long shot.
Nashville Metro schools, with a few exceptions, don't have a desire to be great at sports or education. If your child can leave for a private, one should surely consider it.