Austin and Salt Lake City are about 1,074 miles apart by road. Split the difference and you arrive near Albuquerque, the city closest to the halfway point between them. That puts roughly 12 hr 54 min of driving on the Austin side and 10 hr 10 min on the Salt Lake City side — the fairest single meeting point among the cities near the middle.
Over this distance most people will fly rather than drive the whole way. Albuquerque still makes a fair, central place for Austin and Salt Lake City to converge, splitting the travel instead of asking one side to cross the country.
If Albuquerque doesn't have what you're after, El Paso and Denver are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Austin and Salt Lake City.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.