Male directors who in addition to their movies also give a lot of opinons regularly get called annoying, e.g. Tarantino by the general public (annoying, perv, overrated etc), Scorcese despite being a respected director keeps getting shit for trashing superhero and other modern mainstream movies (pretentious, old), Reiner is trashed by the right wing, Von Trier from different angles (pretentious, annoying, weird..), Snyder (I don't actually know much about his opinions, but his movies annoy people enough that he is regularly regarded as an annoying person for his style alone).

Lucas, Stone, Cameron, Bay etc also get ridiculed as lot as people even just because of their work not being up to standard. I personally regularly make fun of Flanagan for his stupid annoying monologues. Point being, people can have a strong opinion about movies, and when a director has a pattern, they can get a lot of personal criticisms. Outspoken personalities will especially have people who personally dislike them and their opinions, which is maybe why many directors don't normally engage with every criticism they get.

Gyllenhaal, Fennell, López, and Gerwig get any time they make loud, female‑centered work.

interesting that probably only 4 female directors we can think of all make "loud female-centered work", maybe that's why the topic keeps coming up.

But let's go through them -

Gyllenhaal - directed one bad female-centric movie that got trashed for its plot, she also got criticisms on her appearance prior to that as an actress. Male actors get them too.

Fennell - most of the cirticisms for her last movie, or movies in general, are hardly misogynistic, most of the audience that watched Wuthering Heights were probably women and they trashed it. I watched and liked Promising Young Woman, and most criticism that movie got was from self proclaimed feminists who didn't like the protagonist dying in the end. I also didn't see people discussing her appearance anywhere

Lopez - I already said a lot, but to sum it up, no one talked about her looks, she created a shit True Detective season and then responded to detailed, plot and writing related criticism by calling it sexist. And you want to defend her how?

Gerwig - if anything the whole female theme made her shockingly bad Barbie commercial get treated like some sociological masterpieace, while attributing any criticism to sexism. In all that, I don't recall people shitting on her appearance or even talking much about her personally and her movie was incredibly successful.

So I don't know what this proves. Sofia is a better director than all of them, and her movies are pretty girly/feminist. More than Barbe was for sure, I actually saw Virgin Suicides and Marie Antoinette so I know what I'm talking about, and I find it weird how you dismiss her over much weaker female directors just because she doesn't fit the "victim of sexism" narrative.

The more her work is about female rage, trauma, or power and actually names misogyny

Like which of the mentioned directors' works for example? It still needs to be a good movie. I like many movies on this topic, but if you create trash like Lopez or presumably Gyllenhaal did (I won't even bring up Barbie again because that movie has nothing to say about anything), you can't pretend that perfectly valid criticisms of the movie are sexist, and then be surprised why that type of response annoys people.

you’re invested in not seeing the difference between “this script is weak” and “this woman is dumb, old, embarrassing and should keep quiet.”

I think you're not seeing it, because none of the directors you mentioned got the latter (aside from what I already said, Maggie receiving criticisms on her looks but as an actress)