I think you’re seeing things in a different light than I am, and in your response to Tom later it sounds like maybe you’re still getting to the same end point, so maybe there’s nothing worth saying, but I think I disagree with some of this analysis.
Joi’s purpose is to make K feel loved, but we have no insight into Joi’s “mind”, such as it is. She’s literally advertised as “Everything you want to see, everything you want to hear” at least twice. This may not seem like a meaningful distinction, but I don’t think she’s “designed to love K”, she’s designed to make K feel loved. I don’t think she’s a being with agency reigned in with overriding instruction to love her owner, I think she’s a cleverly designed simulation that adapts to display behavior making the owner think something is deciding to love them.
Again, we don’t know what/if she’s thinking, but I keep coming back to that everything you want to see, everything you want to hear slogan. K wanted to be unique and important, so she encouraged that fantasy. Joi didn’t “believe” anything about K, she just recognized an avenue to feed what he wanted to think about himself.
Supporting that fantasy involved helping K when he goes on the run, but destroying her backup isn’t ultimately anything but potentially self-destructive—not an act of defiance that makes me think she had any autonomy or potential to act or grow beyond her programming.
As for K, I don’t remember anything in the movie that specifically indicates the memories he has were given to him (or potentially others) with the intent of uncovering the secret of Deckard and Rachel’s child. If there’s evidence I’ve forgotten I suppose it’s possible, but it took a larger coincidence than any plan I could imagine would account for to start K off in that direction, so I personally doubt it.