(or "Keeping the Important Flaws")
One doesn't generally think of early Silver Age stories as "character-driven"; during the era when Mort Weisinger edited the Superman comics with an iron fist, the emphasis was on fast-paced energetic stories that sold, and plot logic seemed to take a backseat to that, let alone character development. Yet despite the surface simplicity of these stories, there's actually a really strong character-driven storytelling engine to the classic adventures of the Silver Age Supergirl, one that the modern-day 'Supergirl' series could take lessons from.
The backstory for Supergirl is pretty straightforward (if a bit loopy); she comes from Argo City, a chunk of Krypton that managed to ride out the destruction of the planet relatively intact. The inhabitants built a dome to keep in the atmosphere, used lead shielding to keep from being poisoned by the ground they walked on (which had, after all, turned to Kryptonite), and used Kryptonian technology to generate food and other necessities. Kara Zor-El was born on Argo City, but when she was fifteen, tragedy struck--meteor showers shredded the lead lining, contaminating the environment, and dooming the survivors of Krypton to a slow death. They used their remaining resources to build a rocket for Kara, and shot her towards Earth, where they'd learned that one other Kryptonian survived.
The backstory is important to Supergirl, because while they don't harp on the point very often, Supergirl is a very different kind of survivor of Krypton from Superman. She's young (fifteen when she arrives on Earth), and unlike Superman, she was orphaned recently; Superman was a baby when he was rocketed away from Krypton, but Supergirl understood perfectly what was happening when she watched Argo City perish. She's lost everything of her home, her family, and her entire existence except for one person--her cousin, Kal. Superman becomes, in essence, a surrogate father figure, a substitute for her homeworld, and in general every possible symbol of authority a person could have, all wrapped up in one person...who's Superman, on top of that. Supergirl's whole character revolves around being afraid of disappointing her idol, and around trying to be the best person she can be to make him proud. (This also explains the occasional creepy romantic subtext to a few stories; anyone in that situation would probably develop at least a bit of an Electra complex. After all, if you're always comparing your romantic interests to your father figure, who's going to look good compared to Superman?)
Again, it's important to note that they don't make a big deal of this. Whenever a story brings up a character trait, the reader automatically assumes that said character trait will be important by the story's end. If, for example, I write about a character who's got a fear of snakes, the audience can pretty much take it as a given that at some point, that character will be trapped in a room full of snakes, and will overcome his fear in dramatic fashion to demonstrate his growth as a person. This is the sort of thing that readers talk about when they say they want to see "character growth" in comics; people overcoming their prominent negative character traits.
But in the real world, we don't always do that. In fact, it can be argued that we very rarely do. Sure, we'll work on curing something like a phobia or a bad habit, but in general, when it comes to something as deeply rooted as Supergirl's insecurities and her need for approval, we don't "overcome" them, we find ways to make positive use of them. So does Supergirl; she tries to be a good person, and a good super-hero, because at every turn she can imagine how it would feel if Clark was disappointed in her. (Which isn't to say that Clark is disappointed in her; there are a few stories where he expresses disappointment in specific things she's done, but on the whole, he's pretty supportive...for a guy who stuck his only living relative in an orphanage for a year or two while she learned how to use her super-powers.)
This character-driven engine has two major advantages; one, it automatically makes Supergirl a sympathetic character. Insecure characters always gain the sympathy of the audience quickly, because we notice the good things about them before they do; we believe them to be a good person who just needs to realize that. (Humility is always a sympathetic character trait, and insecurity is just an extreme form of humility.) Of course, eventually we expect them to overcome their insecurities, which is why (again, I can't stress this enough) it's important not to harp on how insecure Supergirl is. If it's not introduced as a plot point, the reader won't expect a plot payoff later on.
Two, it's open-ended. Supergirl's original status quo is a false one; she's staying in the orphanage while Superman teaches her how to use her powers, conceal her secret identity, familiarize herself with Earth culture, and be an effective super-hero. But there's only so long that writers can find good reasons why Supergirl should be kept a secret. In the end, they have to reveal her (and they do, in a surprisingly moving story.) But because Supergirl's need for Superman's approval is psychological, it doesn't "go away" when Superman tells the world how proud of her he is. All that means, to Supergirl, is that if she fails her cousin now, the whole world will know about it. Superman's praise means something to her, of course, but she's motivated by wanting to get more of it. That need isn't ever going to be "fulfilled" permanently. Which means she's always working, always striving, always trying to be the best Supergirl she can be. Which, in turn, drives her stories.
By contrast, the post-Crisis Supergirl (the post-Crisis Kara Zor-El, that is, not the 1990s version that was actually a shapeshifter from a pocket universe) is directionless. She never had to earn the right to call herself "Supergirl", she never cared about Superman's approval, and she already believes herself to be super-powerful, faster, smarter, stronger, better, and generally just amazing. She doesn't have the motivation to become a better person or a better super-hero, and any attempts to make her actually heroic tend to come off as authorial fiat, rather than rooted within the character. By taking away her flaws, they've actually made her a more flawed character.