I don’t understand this. I think the reason you’re on here is that your conscience is being pricked. You know that what Emma has done to Anna is seriously out of line. It’s just you desperately want to find mitigation so you can feel some justification in carrying in as you are, and so you don’t have to admit there is wrongdoing. Because then you might have to consider rocking the boat, which is inconvenient for you.
Emma can fall out with Anna- she can end the friendship at any time, for any reason. Her reasons, as you have laid out, seem rather tawdry- Anna disagreeing a little too vehemently on a political opinion; Anna taking Emma’s husband to task for behaviour towards her sister which you have described as horrible and misogynistic (which was done privately, when he had sobered up- seems entirely reasonable to me). However, it is Emma’s prerogative to decide to end a friendship over that.
But- Emma didn’t just withdraw from a friendship, she mounted a prolonged, pointed attack by freezing Anna out and getting everyone in your group to join in and thus isolating her from her friends. She did this knowing Anna was having a difficult time health wise and was vulnerable. She then publicly humiliated her. That is utterly appalling and totally indefensible, no matter how nice, fun and supportive Emma might be if you are in her good books.
If your friends were just not talking, i could kind of get keeping out of it and letting them sort it out. But the minute it turned into vicious bullying of Anna- your supposed friend- that strategy became cowardice.
You let your friend be publicly bullied, for what seem like very petty reasons, at a time she is vulnerable. You supported her in private, under cover when you couldn’t be seen. That’s cowardice, of a sort, too. It’s like the man who dates a woman he genuinely likes/finds attractive but won’t be seen in public with her as she doesn’t meet typical beauty norms- maybe she’s overweight; unfashionable or a bit of a nerd; from a different social, ethnic or religious background. He’s afraid to admit who he likes publicly because he fears what people might think and he’s ashamed/embarrassed. Your the equivalent, offering support just as long as it’s quiet and doesn’t openly contradict Emma’s position.
At the point Emma decided Anna could not be a part of the group because she, and she alone, decreed it to be so and you stood back and let it happen with nary a question it was cowardice.
You contradict yourself by saying you are too frightened of the backlash you’d face to stand up to Emma. But then say she’d never be horrible to you because she’s have too much to lose socially. Which is it? I think the former, and you enjoy being in the reflected glory of Emma and her “amazing” husband, and don’t want to be shut out. If Anna was the price to pay for that? Well, it’s a price you chose to pay. I just hope you are right and your wonderful dinner parties and important social connections are enough to keep you where you want to be. Maybe it will, but it’s also possible someone with better parties and more prestigious social connections might come along- and if your face no longer fits you might end up on the other end of Emma’s less lovely aspects. I’ve seen it happen, don’t kid yourself it can’t.
Honestly, I think you’ve really let Anna down. Unless you really mean it, and will deal with Emma differently in the future, leave her be.