J.H.
I think if you are wanting shops, restaurants and public transportation you should be considering living in Austin proper. Especially if you don't have to worry about schools.
Tarrytown is very nice, as is the Belmont, Pemberton Heights and Clarksville neighborhoods. Also some nice areas near Zilker Park.
It is a fantastic town (though beware - summer is HOT). I went to grad school there and my sister currently lives there - she lives in the Belmont neighborhood and loves it. It's my favorite Texas town.
Take a weekend trip there and hook up with a realtor that handles houses in one of those neighborhoods. Or just explore the areas on your own.
Is this where you found the "South Central area" designation?
https://www.mapsofaustin.com/austin-neighborhoods
I'd never seen this site until I was looking up something for this answer, and the neighborhood maps are wonderful.
One place you might want to shy away from is East Austin (east of I-35). It was bad when I was there, but that was 20+ years ago. I know many people have been moving over there because of cheaper house prices, but it is probably a little more "urban pioneering" than what you are looking for.
Good luck!