I really dislike the way you’re portraying feminism as a brand and trying to assign responsibility onto individuals for the public perception of that brand

Feminism is a brand in the same way civil rights are. There's a reason why MLK succeeded where Malcolm X failed, Gandhi successfully took back India, Obama won the 2008 election, etc. This all has to do with how they're perceived to people not part of their movement. Without a good brand none of these movements would have ever succeeded. And yes it is up to the leaders and each individual member of these movements to uphold a generally good perception. Thinking otherwise is ridiculous. You have to win over the population, always.

It’s not the responsibility of any woman to convince men that they deserve rights, that they deserve fair political power and representation. If someone is dissuaded from supporting women’s rights because someone said something they didn’t like or agree with, that person is a misogynist and unlikely to have ever actually supported women’s rights in any meaningful capacity.

In an ideal world no, but we are not in an ideal world. If someone is a mysgonist what is so wrong with sitting down with them and discussing topics like normal human beings and showing them why that's wrong? Just completely shutting them out like how you're describing is exactly how you embolden an opposition group. Imagine someone on twitter was actually just simple minded and based their opinions on one tweet and didn't actually hear the other side properly? A lot of people like that exist. And if your attitude is "oh they're misogynistic and never cared so I shouldn't even bother" then you're just digging your own hole.

The caricature of the “crazy feminist” is also in and of itself misogynistic, and is used to silence feminist activism all the time. Not that there aren’t legitimate extremist parts to the movement, particularly in the 60s 70s and 80s when feminism had yet to make many major strides towards female liberation. Just that the label is often used to dismiss things like the pink tax, the wage gap, and discussions of rape culture and intersectionality.

See what I, and I'm sure many others dislike is the way you derive misogyny from a simple example. A lot of people simply don't see anything wrong with calling out the "crazies" of a group. Am I islamaphobic for calling out terrorists? No. Am I anti-christian for calling out the Westboro Baptist church? No. Am I misogynistic for making fun of clearly unhinged people on twitter? No. Extreme examples of course, but you get the picture. The instant jump to misogyny when genuinely crazy, unhinged, insane feminists get made fun of is ridiculous. Like I said, >99% of feminists are completely normal and sane. There is nothing wrong or hateful for calling out the crazy people in any group.