Privilege plays a huge role in the classroom. Let me give you a few examples.
First, privilege gives you the best of the things of the world to give you an advantage. For example, if you need a tutor, you can afford one. If you need to travel to experience the world, money can do this for you. In addition, privilege gives a family leisure. This is a huge benefit as well.
Second, privilege also gives social capital or a certain frame of mind that allows a person to get ahead. In other words, a person with some means knows how to carry themselves better. These social graces are important in getting ahead in the world and in the classroom.
Finally, privilege comes with many connections in life. As the adage goes, "it is who you know; not what you know." This is an enormous benefit.
I will add the social theory of Pierre Bourdieu. He writes of social capital in profound ways. It is worth a close read.