I think you're making a false equivalency here. Both sides are very thin-skinned, no doubt, but one is a direct insult ("you are deplorables") and one is taking general offense on behalf of other people for perceived slights.
I think you're right. It's not really about insults it's about free speech. If you say something white supremacist like "our company hires mostly white people because there are fewer qualified people of color in the workforce" and I say "you're a white supremacist", what will happen?
1) you'll accuse me of name calling, even though I just gave an accurate description of your political platform
2) you'll accuse me of trying to curtail your free speech, even though I was just exercising mine
3) you'll try to invalidate what I'm saying by invoking the "you may not campaign for the rights of others" rule you cited
All of which just serves you avoiding actually addressing my words, which are clear and direct: you are a white supremacist.
If I'm lucky you'll try to defend yourself by rattling through the list of your racist beliefs and then we can have a real discussion.
But most anti-SJW types are so triggered by words like "racism" and "xenophobia" that we never get that far.
(I voted for Hillary, mind.)