Okay, gonna throw my two cents in here.
The show wasn't terrible, for my part. It wasn't "Wonder Woman Pilot" bad, it was just generic girl power fluff. They did some decent nods to the comics/fandom (Cat Grant, the casting of Kara's adoptive parents), and had several elements that felt very much Silver Age, which I think is a good thing for DC properties; a dash of tongue-in-cheek.
Now, I don't disagree that the quotes in the OP are absolutely cringe-inducing. The script is stilted, and the exposition is ham-fisted. The fact that they never really mention Superman (only calling him 'my cousin', Kal, or things like 'the blue guy'), and yet rely on him to establish the universe, struck me as a little too "Birds of Prey" if you know what I mean*.
It was definitely worse than the Flash pilot; it relies way too much on you being already invested in the idea of a Supergirl show, so it barely develops any of its characters; and when it tries it's generally delivered as clunky "twists" or through the character blankly describing themselves to you (I'm look at you Cat). That said, it isn't worthless or horrible, it's just meh. She does have to go through conflicts (though they are ridiculously forced to try to enforce the "her weakness is her emotions" idea), and they're setting up for a major conflict (which, honestly, Superman should be appearing to help with, but I doubt will; diluting the property and all that).
My biggest low point was the way they handled Jimmy Olsen. In the Silver Age, he was a romantic interest for Supergirl, and they're clearly setting up at least an attraction for her here. However, in the comics, it was interesting because she saw through his awkwardness to find someone who was genuinely a good person, and here it's because he's a hot guy. Add to that they're writing him as a confident, mentor figure, and I don't really see the point of calling him Jimmy; oh, I'm sorry, "James".
But yea. Not terrible, just not really for me. It's really being aimed at Tumblrinas who don't so much read or know the comics, but are aware of her through image-posts. That said, they are clearly doing nods to the fandom, so maybe this "PowerPuff Girls as written by Dawson's Creek" is just a way to get more people to watch, and will fade as the show progresses. I won't know; the pilot killed any curiosity I had in the show's development. I'll likely keep track of it like I do Arrow; through reviews.
But yea. That's my stupidly long ramble. TL;DR: It isn't bad, and has some small potential, but the Pilot is relying on "Grrl Power" more than the source material for ideas.
*- Birds of Prey was another attempt at a Girl Superhero show, but this one starred the Huntress as the daughter of Batman and Catwoman, while working with Oracle (Barbara Gordon) and the daughter of Black Canary. Her mother was killed by Joker, and her father disappeared in a final battle with him in a warehouse. It never once said Batman, only "Bruce" or "my father" or "Mr. Wayne." They tried as hard as they could to keep it from connecting to Batman, while actually having Oracle don her Batgirl gear, and have Harleen Quinzel be the chief admin at Arkham. It was a decent show, with a slow start, that suffered from WB/DC trying as hard as possible not to "dilute the brand" of Batman by allowing the show about his daughter to be directly linked with it.