I live in the Belle Meade area of Nashville. Moved here from NYC in 2006. Before that lived in Austin. Grew up in East Texas
I'm biased but think Nashville checks more of your boxes than you think. Like Texas, no state income tax. Percy Warner park has tons of trails for hiking, horseback and mountain biking. There are a lot of state parks within an hour or so, particularly on the Plateau that offer tons of hiking.
The food/entertainment scenes are great, as you mention. Remodeled airport opened this year. Direct flights to London now and I'm told Paris and Tokyo soon (Tokyo, because Nissan and Bridgestone's headquarters are in Nashville)
I don't spend much time in the Franklin/Williamson county, so I don't see much of the MAGA nonsense, but I know it is heavy there.
Seems to me that the Belmont/12th South areas, as you mention, might be good spots to look. I would also throw out my neighborhood and Forest Hills.