NOT saying that Emma has no faults, but disagree with your harsh analysis of it.

>Emma did care about being accomplished

Of course she did -but not enough.

Mr Knightley is like parents who push their child towards perfection. He wants a six. 4 is not enough, or even 5. At 37 he wants to bully Emma to be 37 at 21.
He accepts it later. Only 'he' calls it 'blame' as the word 'bully' didn't exist then, I think.

"She did unfeignedly and unequivocally regret the inferiority of her own playing and singing. She did most heartily grieve over the idleness of her childhood -- and sat down and practised vigorously an hour and a half.

LOL!!! Don't you see humour here? I can imagine JA's tongue firmly in her cheek, describing Emma's feelings with words like 'unfeignedly' 'unequivocally' 'heartily grieved' and practised vigorously
Emma won't feel this later, nor will she practise vigorously. :-D

This passage describes Emma very well. Even her envy (not jealousy) of Jane is not 'all consuming' as many would have one believe. It's not eating her up. It's NOT the dominant thought 24/7.

A jealous person would find fault with Jane's playing anyway. But Emma says in CH27;
"My playing is no more like her's than a lamp is like sunshine"

..AND she says this to Harriet who goes on flattering Emma.
"Every body last night said how well you played."

To which Emma responds;

"Those who knew any thing about it, must have felt the difference.
She praises Jane further;

The truth is, Harriet, that my playing is just good enough to be praised, but Jane Fairfax's is much beyond it."

Undeterred - Harriet continues to flatter Emma;
"Mr. Frank Churchill talked a great deal about your taste, and that he valued taste much more than execution."

To which Emma responds;

"Ah! but Jane Fairfax has them both, Harriet."

This generous praise is not a sign of petty jealousy.

And Mr Knightley thought Emma would get more spoiled with Harriet's flattery. Mr Knightley surely underestimates Emma a lot. He's very very harsh about Harriet too.
After saying - so harshly;
"She knows nothing herself,

He later accepts at Crown Hill ball;
"Harriet Smith has some first-rate qualities,"