In addition to having to pay hundreds of millions of dollars to build new launch facilities in South Texas, you'd still have huge range safety issues.

The launch azimuth you use depends on the launch site latitude and the desired inclination. The minimum inclination you can directly launch into is equal to the launch site's latitude, and you get that by launching due east (or west, but retrograde launches have a lot of drawbacks). The formula is quite simple:

acos(inclination) = cos(latitude) * sin(launch azimuth)

where angles are in degrees.

Examples:

A launch due east from the Cape:

acos(inclination) = cos(28.5) * sin(90)
<tab><tab> = acos(0.879 * 1.0)
<tab><tab> = 28.5 degrees inclination.

A launch from the Cape on an azimuth of 60 degrees results in an inclination of 40.44 degrees. Likewise, a launch azimuth of 110 degrees results in an inclination of 34.32 degrees. Higher inclinations are much less desireable for interplanetary missions and even more so for geosynchronous satellites.

As for range safety, in case you didn't notice, rockets drop pieces as they fly, either intentionally with staging or unintentionally with explosions. That's why they avoid overflying populated areas during the boost phase. Take a look at a globe. Depending on the inclination you're aiming for, you'll find a lot of populated land (Florida and Cuba for starters) to the east and southeast of Texas. There's also the Bahamas and a lot of other populated islands that would limit launch azimuths.