Thinking back, I think this episode explained more than anything why Kara would consider Alex her home but not her adoptive parents.
Eliza (and Jeremiah) were always great to her, but they probably reminded her too much of what she had lost; of her parents. Add to that any guilt she might feel about "betraying" her parents by showing even the slightest amount of love to a new set (even though Eliza has said that she doesn't intend to replaced Kara's parents) and it is understandable why she would close herself off from Eliza and Jeremiah, while still appreciating their support.
The rest of human culture must have seemed something that was pushed onto her. Especially since she expected to live with Clark and take care of Clark and instead Clark left her with these new folks she didn't know; and now everyone is asking her to be human. This is a girl who lost everything: not just her parents and her friends, but her planet and her culture. She must have wanted to hang onto whatever she had of her culture as tightly as possible and reject human culture so that she can keep Krypton sacred.
Hence the need to reject school and just hang out with Clark and be Super. Clark after all is her only remaining connection to Krypton; and someone she knew (as much as you can know a baby) and loved in Kryton. She is a teen and probably doesn't understand that Clark here is not actually a kryptonian; but a human in Kryptonian skin and that they would have as much of a cultural gap as her and Eliza. And she would doing what her mother said she would do (be a hero) rather than whatever weird earth things her new guardians are asking her to do.
She was actively seeking and trying to save Krypton in her mind while rejecting Earth.
But Alex. Alex was the only thing that didn't remind her of Krypton or her loss--because she was someone new and just herself, a new relationship that Kara has never had; and was too self absorbed--like most teens are I suppose--to actually cater to Kara's loss.
As the only thing outside of Krypton, Alex probably intrigued Kara at the beginning. And, she could feel with Alex the way she couldn't feel with any of the others--Eliza, Jeremiah, even Clark and of course her high school folks. So, she probably paid a lot of attention to Alex, not the least because Alex was "saddled" with babysitting her through high school.
All that little competition, "try and keep up" and fighting with Alex was not I think Kara just being Kryptonian Kara; but Kara imitating Alex and trying to meet Alex in her own terms. It is what I have seen younger siblings do when they don't get the elder's attention (which is pretty much always). Or, at least, that is what my sister did. She used to just follow me around and then irritate me by doing stuff I hated and fight with me. And I was pretty much always trying to leave her behind (I was a horrible elder sister in my teens).
In that discussion with Kenny at the school cafeteria, it is pretty clear Kara wants Alex to notice her and appreciate her despite that talk about invisibility.
Hanging out with Kenny might have helped, but he was still a stranger.
And then, Kenny died, and Kara had Alex spending time with her and actually appreciating her for her abilities and finally trusting her to save her. Probably the first time Kara ever felt happy with her life. The first time ever she felt at home and as if she belonged after she lost Krypton and her parents.
That feeling must have continued and built on as the years progressed. Pretty solidly given the ending.
As for Alex: She was also feeling a loss (much less catastrophic than Kara but bad enough in Earth terms). She lost life as she knew it, her parent's attention (quite bad for a 14-year old) and then, her father whom she worshiped. And since the first two happened because of Kara why not blame her for all three. Easier to do than dealing with the loss and moving on.
Even without that, having take care of a younger sibling all the time is irritating for a teenager.
Having to then face rejection from friends she hoped to count on and getting support from the one whom she had been rejecting all this while may have shown her where her priorities should lie. (Not that she was all set against her; she did look out for Kara even before; by chasing away bullies at the lunch line). The Kenny episode showed her Kara is the one person she can always count on; and the one person with whose support she can get through anything.
Leading us to the closing scene with the Danvers on the balcony. I think that scene put a nice punctuation on the entire story by emphasizing both what they mean to each other and why they mean so much to each other.
It doesn't mean (and I know some folks are looking askance at this) that all their issues are resolved. It just means that they have remembered what they still have and are counting their blessings. The moment of silence, retreat and simpler times made them remember that the world has not still ended. And that they can still count on each other.
Doesn't mean the conversations are over either. They are just not ready to speak about things yet. But, that is okay. For the moment, silence is what is needed.
flashback of the season, was more of an interaction between the flashback and the current time period. But, as it was done, it didn't damper the experience. 
