Your argument seems to be "as long as people still commonly use the term, it is okay to keep using the term". Of course this will mean no one will ever need to stop using "woman" as a derogative term. (As an aside, if you want to see my wife pissed off, say something like "you play like a girl" in front of her.)
And had we followed that argument, we would still be saying "stupid black" and "stupid jew".
As far as "getting your point", yes, I absolutely get your point. People are taking something he said and making the most of it. Politics as usual, and typical bullshit. It's wrong.
But there are two separate things you are talking about here: The first is the use of the term "woman" in a negative way, and the second is how politicians (and others) exaggerate or lie to take advantage of someone or something.
The problem I have with your argument is that you seem to be conflating the two. You are going down the path of accepting the use of "woman" as a perjorative in part because what the other side did was even worse. From my perspective both are wrong. I'm not sure why you are conflating the two.