On the one hand, I totally agree, as far as you've gone: Anne indeed very likely is legitimately sickly from some disease that was at the time unpreventable and barely treatable, Lady Catherine is not to blame, and blaming her is sexist and regressive.
On the other hand, Lady Catherine is overbearing to literally everyone she meets - even to Darcy on occasion, and certainly to the Collinses, to Elizabeth and Maria Lucas, and to the Bennets when she visits. It would be unusual if she were not overbearing to Anne.
Why is she overbearing?
Maybe it's this situation. Maybe she has had a string of miscarriages, stillbirths, and dead babies, and is experiencing constant and overwhelming grief - and the one child who survived to adulthood is really sickly, which makes Lady Catherine constantly anxious. Maybe she actually loved Sir Lewis de Burgh - which I think the fandom probably can't imagine, but there is nothing against it in the text - and she misses him terribly. She very likely loved Lady Anne, Darcy's mother, and misses her terribly, too. Maybe she just has a ton of grief over things she could not control.
And maybe she feels guilty about the string of the dead children, which people have been putting on mothers for centuries, millennia even, despite the fact that it could easily have been Sir Lewis's fault - or "fault," since he would also not consciously produce unviable children. It was beyond the control of both of them, but hey - somebody must take the blame, right? "Let's blame mom" is a common and long-standing custom, so she internalizes this cultural message.
So from that painful mix of grief, guilt, and anxiety, she tries to use her very high social position to exert as much control as she can over her world. Personalities are different; maybe this is how her particular personality dealt with these life-reverses. Someone else would respond differently. She goes with overbearing and controlling.
Fandoms trend young, and often haven't had adverse, adult-only experiences (like losing several family members), so they often don't have the compassion required to avoid a sexist and regressive blame-the-mom stance. Also, Austen herself set up Lady Catherine as a figure of fun:
"Elizabeth soon perceived, that though this great lady was not in the commission of the peace for the county, she was a most active magistrate in her own parish, the minutest concerns of which were carried to her by Mr. Collins; and whenever any of the cottagers were disposed to be quarrelsome, discontented, or too poor, she sallied forth into the village to settle their differences, silence their complaints, and scold them into harmony and plenty."
Scolding someone into plenty is obviously satire, so unthinking fans will just go with it and take it too far. Even if it takes some of the fun out of reading Austen, understanding Lady C's point of view (even speculatively, as I did above) might help fans avoid this nasty blame-mom tendency.