inferior to Emma classwise, and that's the thought she's having.

About 200 years ago education was not common, and those without it *were* coarse and unpolished...generally speaking

The problem here is that Emma is applying the generally accepted view, and shows it when she says;

she knew Mr. Knightley thought highly of them -- but they must be coarse and unpolished,

>Emma very clearly thought to elevate Harriet to a higher level in Highbury society--a level that would make her acceptable as an intimate friend of Miss Woodhouse.

Emma 'felt'/was pretty sure that Harriet was a gentleman's daughter and needed help especially because she had
decided to pair her with Mr. Elton, right from the beginning, the first time she met Harriet.

CH:4
It was not likely, however, that any body should have equalled her in the date of the plan, as it had entered her brain during the very first evening of Harriet's coming to Hartfield. The longer she considered it, the greater was her sense of its expediency.

So it wasn't to make her acceptable as her friend.