Your reasoning for why the non-italicized portion is not good enough is “conservatives say that”. The reason you offer for why the professors get a frosty reaction is “they could be mistaken for stealth reactionaries”. Perhaps it’s less important to consider what label you can assign to someone and more important to just consider what they’re saying.
And it's absolutely relevant what a person "is." If NRA TV uploaded a special report on gun control, it matters that you know it was produced by NRA TV. I can't believe I'm even having to argue this point.
Finally, I didn't defend students/academia reacting incorrectly to the professors, I just said I understood. A lot of people are (rightfully, I think) sick of neoliberalism's "free marketplace of ideas" approach to permitting right-wing toxicity, and can get carried away when they think that's what a speaker is relying on. That doesn't make them automatically right.
Personally I think that who someone is is absolutely useful information because it can contextualize the intent of an argument and provide a hint as to the biases of the person making the argument, which is a useful tool for thinking critically about the argument. However, when weighing the merit of the argument, it doesn’t matter who is making the argument.
That’s why I took issue - you said that the reason the italicized portion was needed was because without it you sound a lot like a conservative. That’s weighing the merit of the argument based on what kind of person makes it rather than the substance of the argument itself.