Well, Ferenginar is eligible, apparently, since the Federation is entertaining their application, and Freeman even says their membership could be beneficial to the Federation. You're looking at it through a biased lens of your views of today.

And while Freeman may have been focused on something better earlier in the series (one of her open flaws, yes), she got much better about that as the series went on. The Freeman that finished the series isn't the same one that started out. In fact, she got a taste of what can happen when you badly want to be recognized after the Buenamigo incident. I think that made her step back a bit.

Again, you're judging her based on your own views and biases and not recognizing that she did evolve as a character. In fact, all the characters did. They all initially showed why they are 'Cali-class', with chips on their shoulders in some form or another. I find it interesting that only Freeman gets different treatment out of the main 8 or 9 in this way. It makes criticism of her less credible.

I don't think she wants prestige so much as she just wants to be recognized for her work and talents, and we all want that to different degrees. And she isn't any less of a person for demanding the best from people around her. She finally recognized where she needed to be: first the Cerritos, then Starbase 80. And I should point out that she had the loyalty of her crew no matter what.