I agree with you Elinor and Marieanne come to appreciate Mrs Jennings for herself.
Elinor always accepts Mrs Jennings despite her faults.
As the narrative unfolds, they remain aware Mrs Jennnings has her brash, gossipy faults yet they respond to her warmth and good nature.
As the narrator describes Mrs Jennings as 'rather vulgar' I don't see why the Miss Dashwoods should be expected to deny it.
At least they come to see Mrs Jennings coarser qualities are outweighed by her great kindness- that is not elitist.
John's reasons for wanting to meet Mrs Jennings are different. 'I understand she is a woman of very good fortune'. When he sees Col. Brandon, John 'only wanted to know him to be rich to be equally civil to him.'
Snobbishly, John is civil only toward rich or titled people- an elitist attitude different to his sisters.
They did not seek favour with Mrs Jennings, she took a fancy to Elinor. Nor do they don't attempt to ingratiate themselves with Lady Middleton.
There are more differences between the attitudes of John and his sisters than any similiarites.