I think the parent is on to something about the implications of the argument, although it is not how the post's author would look at it, of course.
To really understand an argument, one has to start with its subject. Are we talking about viewpoints or people? What the blog post author focuses on is people:
> Me: Now lets try it for politics. How many of you would say you are on the right politically, or that you are conservative or Republican?
This is a fundamental misunderstanding of how discourse is conducted and should be conducted. People are not on the right politically, conservative, or Republican. People are people, and each might hold or articulate viewpoints or opinions that are on the right politically or conservative. They might be a member of the Republican Party. All of these are elective and potentially temporary.
The whole point of discourse is that viewpoints, opinions, and memberships can change. So to start by framing those as elements of identity needing protection, feeds directly into a framework for discussion that is conflicted. If a discussion must validate and protect all viewpoints (conservative or liberal), then what is going to be discussed?
IMO the right way to approach this situation is to explore the potential consequences of voicing an unpopular opinion. Often, they are far less scary than teenagers might suppose. We should focus on how to give each individual student the mental tools to effectively evaluate arguments, and to manage their anxiety about going against perceived social norms.
Teaching kids to be brave in speaking up against prevailing opinion can help create positive outcomes throughout their lives. We want citizens who will speak up for what they know is right, even if they know they will face trouble.