I've found it to be both enlightening and effective in these kinds of conversations to make sure to ask very simple questions about what we're really talking about. Why? Because for example I often have people say to my face that the american democrat party is a communist party. Hah. Or, more grim: that queer people are groomers. Terrifying.
So it's an entirely fair question to ask you what you mean by "cancelled." I genuinely don't know. You said a professor was cancelled - but the professor was able to give a speech at the same university. I don't understand! I really don't! What does "cancelled" really mean? Because if you can't answer that, maybe this thing you fear doesn't actually exist. Maybe you really are just functioning as a reactionary enforcer of status quo and lashing out against actual counter-cultural speech, against arguments that are actually being "cancelled" by you and those that support your position.
The same for gender. You said to me on another comment, that "you aren't talking about gender identity," but you earlier in the same chain said "care givers tend to be women." Yet it's as obvious to me that that is a circular definition of a gender completely detached from biological markers ("women are caregivers") as it is to you that men are big masculine lumberjacks. So instead of letting you get away with, I don't know, some kind of assumptive bias? Begging the question? Argument to tradition? Instead I'm simply asking you to demonstrate why it's acceptable to assume "care givers tend to be women." Personally as a man I reject this because it leads to pretty shitty outcomes like me being given stink eye from people when I'm "alone" at a children's playground... until my kid runs up to me. To give a concrete example.
Not to mention I reject that "race car drivers tend to be men" for the obvious nod towards the very annoying stereotype that women are worse drivers than men. Ugh. I'll give you enough good faith to assume that was unintentional on your part.
TLDR you presented a presumptive conservative position and got reactionary when asked, by different people (annoyance at me is normal, I am an annoying person), very simple questions about terms setting and definitions. So? What's up with your ideas? They don't stand to scrutiny? I mean you're doing what your position tends to accuse my position of, which is for me why I put my foot down so damn hard in conversations like this: you're acting offended, you're responding rudely and, frankly, irrationally, and straight up I don't see any rhetorical, logical, ethical, deductive, or evidential basis for the broad vaguely conservative smear I find your belief system to be.