For sure, Texas and A&M get most, if not all the publicity for the Texas Higher Education system. However, the system is much much larger that that. As 2B mentions Texas Tech, and University of Houston are both good schools. Also there is a tier of schools, Sam Houston State, Stephen F. Austin, North Texas, University El Paso, Arlington, and I am sure a lot more I am not aware of. Your child will get a good education at any of these. Some may actually be better than UT or A&M depending on what they want to do when they get out. All are generally cheaper, as tuition is less and living expenses are also less.
In my experience the main thing a college degree proves is you can get up on your own and make an 8:00 class. If you graduate at the top of you class at any of the major universities, of course, you are in a good place. However, those that just graduate, IMO, are not at any great advantage compared to those that graduate high from one of the second tier schools. All can usually get in to graduate school and most will get jobs in their profession, (if there are jobs available).
So, if they just want to go to school in Texas, there are plenty of inexpensive choices that will lead to economic opportunity. By the way, Rice is expensive, but their original charter did not allow them to charge tuition. I think it was back in the 70's that they changed it, and when they did, the change provided for very liberal scholarships based on need. A real libral attitude 'Those that can afford it will pay, those than can't will not'. Not sure if that is true today. Might ask Nords, as his daughter just took the tour.