I do think you have a point there.

Emma has often to do with illnes /suffering during the course of the book.

She doesn't mind visiting them (the poor, Harriet) or looking after them, like she did her father.

There is the case of her visiting Harriet when she was unwell, and Mr Elton getting all worked up about it.

She wanted me to nurse my cold by staying at home to-day, and yet will not promise to avoid the danger of catching an ulcerated sore throat herself!

I'm wondering if that was the general way - not visiting the sick in case one caught it.
What surprises me even more is that Mr Woodhouse doesn't stop her from visiting the sick, and he's so afraid.

She even wanted to keep Harriet back at Hartfield so she could nurse her.

....spent by Harriet at Hartfield, and she had gone home so much indisposed with a cold, that, but for her own earnest wish of being nursed by Mrs. Goddard, Emma could not have allowed her to leave the house.

Sometimes I wonder, had Miss Bates been ill, would Emma have been compassionate towards her then?

Coming to Jane. I do think Emma's compassion rose every time she confronted illness, and so it did for Jane - most of the time.

In Jane's case her contradictory feelings are so well sketched out by JA. I think it's a normal reaction (this see saw of feelings) when a persons kernal is good to start with. Doubt Mrs Elton would ever go through this.