Cars are great for many things, but not navigating a dense urban area.
Take a cue from many old European cities that grew before and after the invention of the automobile. They did not have a choice to build more roads downtown. They invested in real transportation - light rail or similar downtown, buses connecting less dense areas to the dense areas, and large parking lots on the outside of town.
We should have light rail connect downtown, the university, S. Congress, the Domain, and eventually other dense areas. It would spur growth even more in these dense areas. Take away half of the streets in dense areas from cars and convert them to light rail and bike/scooter lanes. Let pedestrians have the sidewalks back. Build parking garages outside the dense areas with free parking and paid rail/bus connecting these lots to the dense areas. It would reduce everyone's stress, save time, and in the long run provide a more viable solution than adding more cars to already crowded streets. Visit Amsterdam for inspiration.
We also need a highway that completely bypasses Austin if travelling from San Antonio/Mexico to Dallas. The state should pay for it with trucking fees or a tollway.