There is a little bit more to it than that. NASA is not just a space agency, National Aeronautics and Space Administration.
NASA also does a lot of aviation and other research besides just shooting people into space and playing golf on the moon.
The reason for launching in Florida is because if something goes wrong they can initiate the termination system and anything that falls out of the sky will be in the ocean, opposed to a populated area. Other NASA centers in other states are useful for shuttle returns. Often when bad weather is occurring they send the shuttle to land at NASA Dryden.
It's "odd" how NASA facilities are so often located in states that had politically powerful senators at the relevant time.
The "Robert Byrd" effect is not new. I'm waiting for them to rename West Virginia after him.