It’s sad that it matters, but it just does. Southerners and Texans aren’t going to hold it against her, because they experience the same thing. Upper mid-westerners and Northern Plains folks won’t, because that’s how they sound. Both coasts and D.C. will and do hold it against her, and that’s where virtually all of our media originates.
Holding onto a pronounced regional accent "speaks" to a gratuitous egotism and stubbornness; winners do what it takes to get the job done. It's no moral compromise to have a neutral accent. it may even be a signal that you are not going to favor one group over another.
When I moved away to college, I eased out of my regional accent as soon as possible and adopted what I called "television announcer English." It made going through school easier, and it made doing business easier once I got out of school. Not everyone has an "ear" however for how they may sound to others; that's where a good vocal coach comes in. She needs one.