Insults are always going to based on stereotypes Aron. You cannot change that.
Here, take this less common but still pervasive insult: Stop being such a catholic.
It stereotypes Catholics as being excessively puritanical. And as a Catholic, I don't mind this stereotype one bit. I think it's funny. Yes, we Catholics sometimes take things to an extreme. The insult makes a general statement about Catholics that simply doesn't hold true in many cases. It ignores a lot of nuances and is just based on a stereotype, but I don't think using that insult makes you a bigot.
What I'm trying to say is all insults that target a specific demographic are by definition bigoted, but using them doesn't automatically make you one.
So yeah, "dickhead" and "pussy" are sexist insults, but using them does not necessarily make you a sexist.
Heck, sometimes these turns of phrases make no sense. Have you ever heard "let's go dutch" at a restaurant? Using that turn of phrase doesn't mean you hate dutch people. I don't even know where it came from.