Doesn't everyone have some right to not warm up to someone else? Can you really regulate who is friends with whom based on a social obligation to be good to those less fortunate?

Emma is always civil to Jane Fairfax and only lets her own irritations show when she's with people she trusts not to make those irritations generally known. Granted, she ought to be more discreet than she is.

Put yourself in the position of being the best student in the class until someone new moves to town. Everyone you know fawns over the newcomer. They expect you to be best friends because you're both smart--even though you have little else in common. And then that newcomer doesn't make any move to be close to you, but rather shuts you down any time you try to be chatty and more intimate. In other words, Jane isn't making a lot of effort to be friendly with Emma, either. And Emma's feeling like Jane won't meet her half way has a very dampening effect.

I don't think Emma really dislikes Jane. She feels that other people have expectations about their friendship and her admiration of Jane that she can't live up to, and it makes her cross where Jane is concerned.

Don't get me wrong--I know that Emma is seriously flawed. But I think there are two people involved in the relationship not working.