Which schools offer the best opportunity to become an Astronautical Engineer?

So I have several schools in mind: UT Austin, Texas A&M at College Station, UC Berkeley, and UCLA. Although Rice, (and even UC Berkeley) don’t have a specific Aerospace Engineering major, they are schools I’d really like to attend, so is it a smart idea to major in mechanical engineering and specialize in aerospace? Or is it way better to major specifically in aerospace engineer at UT Austin or A&M?
Also, I have about a 2080 on the Old SAT, and I’d like to possibly work for NASA one day. (Which I heard from a NASA education employee an aerospace degree would limit your job opportunities there, while a mechanical degree would broaden them.)

Yes it is. Companies hire both. If you’re interested in NASA also look at UA Huntsville.