]"They were of course very anxious to see a person on whom so much of their comfort at Barton must depend;"

I don't think this means that Lady Middleton is particularly kind, or actually anything about her at all, since that desire was in expectation of meeting her for the first time, and based on hope alone--nothing to do with Lady Middleton's actual personality. Natural taste cannot be attributed to her before they have even laid eyes on her!

The line means that if Lady Middleton would prove to be a kind and generous landlady or patroness or whatever is the appropriate term, their situation and comfort at the cottage would be much better.

It's clear that she has good breeding in the sense that she knows the socially proper and acceptable way to invite people, accept invitations, etc. but I don't see her as being particularly kind in any way. In fact, they find her cold.