This usage of face is figurative. It comes from the idea of face being the facade/outward appearance of something. As [MacMillan][1] defines:

> the qualities that something such as an organization has, or wants people to think it has   
*This is the new face of banking in America.*

[Cambridge][2] also has a page here:

> what you can see of something or what shows:  
_Poor quality is the unacceptable face of increased productivity._

Basically, according her father, the late girl, Tibbetts, would not want to be the thing that people remember the debate by. This sentence is in the subjunctive mood, saying that if the girl were alive, she would hate it if people saw her as typifying or symbolizing the debate about the U.S. immigration system.

Yes, it is a very common usage. Here's some examples I made up:

> We all see Gandhi as the face of nonviolent resistance.

> Brother Malcolm was the face of the Black Power movement.

  [1]: https://www.macmillandictionary.com/us/dictionary/american/face_1
  [2]: https://dictionary.cambridge.org/us/dictionary/english/face-of-sth