I get not wanting to just retread old ground, but making some of the most wholesome superheroes ever into pathetic cucks (Superman) and drunken messes (Supergirl) is not going to win you any new fans, and will certainly alienate a lot of old fans.
Supergirl always had her issues, that's true, and you could argue that Tony Stark being a drunken mess (in the comics) was a vital part of his backstory, but this is just Gunn being stupid and thinking he's edgy. Even if Supergirl somehow works through her issues in the movie, I doubt most people will care.
There is definitely a place for flawed heroes to overcome their demons or adversity or whatever, and often that makes for a much better character. But Superman is literally the bastion of hope in comics, and seeing him get shit on all throughout his movie was extremely disheartening. Now we see 90% of the Supergirl trailer having her being a drunken mess throughout, so I am not very optimistic that it will somehow make me all warm and fuzzy about her character by the end.
Kara's issue is that she remembers what was lost, whereas Clark has no memories of Krypton. She's also older than Clark, while at the same time being younger (space travel bs, don't ask) and less mature/experienced because by the time she gets to Earth, Clark is already grown and lived a life with his down-to-earth parents. Clark is literally the most important thing in the world to her, while also being a harsh reminder of everything she lost. Sure, in the real world that could drive someone to drink, but this is fucking Supergirl and Superman. They should NOT be depicted the way Gunn is doing so. You can make them vulnerable and uncertain, but they always, ALWAYS come out with their priorities straight in the end. Watering them down and making them "less" is honestly a disservice to the characters, and I hate it. Yeah, we all get bored with the Big Blue Boyscout, but we also know that's how he is, and to change that is wrong.
The best version of Kara is the animated movie, Superman/Batman: Apocalypse. You can see within the first 5 seconds of this clip that Clark is like a big brother to her, not a fucking doormat/dog watcher like Gunn has him. Kara would do anything to defend Clark, and vice versa. Did anyone get the impression that FetalAlcoholGirl would lift a finger for Superman in the last movie? I doubt she'd even sober up for him.