I have re-read the wheel-part of the letter, and it still seems to me that she is more obviously trying to avoid Mrs. Knight losing a personal treasure than to avoid being forced to spin herself.

The phrases

most sincerely do I hope [my writing to refuse] will not be in vain,

and

I cannot endure the idea of her giving away her own wheel,

and

I could never use it with comfort,

and

I was afraid of making it appear less serious matter of feeling than it really is,

are all so indicative of Author Austen trying to forestall a kind person trying to be generous at her own expense, that I simply must disagree with you. Plus why would she have to be circumspect with Cassandra about her reasoning? She certainly shows enough frankness about other things in her letters to her, that sudden formality about Mrs. Knight's possible gift seems a little odd to me.