Do you think that response won't work, isn't worth the effort, is aimed at a non-problem, or other criticism?
I have a lot of time for the sentiment in the blog post you linked to, but don't think privilege is a necessary concept in order to appreciate it. I don't even believe it's the most obvious criticism of the behaviour in question.
By way of analogy, lets say Pat wanders around everywhere with a sword and Chris doesn't wander around everywhere with a sword. If Pat stabs the defenceless Chris in the chest with a sword, you could frame this in the context of power dynamics, and bemoan how Pat has "sword privilege", but this doesn't really get to the core of the problem.
Calling out Pat's sword privilege doesn't offer any explanation as to why Pat has the sword, or why Pat was motivated to stab Chris. It provides us with a narrative for establishing blame and victimhood, but it doesn't actually tell us anything about the underlying situation or how to remedy it, at any level.
Ultimately, I think that a lot of ordinary social injustice arises because no one speaks out loud "Don't do that." Essentially, unwillingness to discuss social rules.
Saying "Parental Abuse is Wrong" is a useless Applause Light for most people.
Saying "It is not normal to be afraid of your parents, and not normal to be unhappy whenever you're at home" is more likely to be effective at creating good change.
...Calling out Pat's sword privilege doesn't offer any explanation as to why Pat has the sword, or why Pat was motivated to
As Multiheaded added, "Personal is Political" stuff like gender relations, etc also may belong here.