For starters, the writing. Series director Mitsuo Fukuda has made a couple of comments about how SEED was supposed to bring Gundam back to it's roots. This does not seem like a controversial comment until he elaborated. He identifed Gundam's roots were shows like Combattler V. 70's super robot show that the original tried it's best to break away from. This guy has also made dismissive comments on anime being serious or an anti-war discussion, saying it's just anime.
This is a guy who believes the hows and the whys of things are unimportant. That is why characters can have an unexplained super mode, or survive instances where they had no way to get out. He's the one put in charge. And the head writer was his wife, who tried to make the show more a drama than a war story (and, to her credit, the first 30-some episodes of that are pretty good though some people find Kira too whiny. It was once Kira got his new Mobile Suit that it became more Super Robot-like and started sucking balls.).
Calling SEED a updated version of the original is just a show of ignorance. The original was meant to be a serious war story (for the time) that chronicled Amuro going from a child to a soldier, while in SEED the Mobile Suits are going to do cool poses before firing weapons while the protagonist stops being a soldier and instead becomes a hero of justice who refuses to kill his enemies. The original went to great lengths to show enemy soldiers as human beings fighting for a cause they believed in, SEED eventually devolves into a bunch of genocidal fucks committing war crime after war crime (insert FPS online multiplayer joke here). Neutrality is mocked by Tomino but championed by Fukuda.
But SEED was a major hit and a sequel was quickly greenlit. The older fanbase was pissed at how SEED got so childish, so in the promotional materials leading into Destiny promised it would be a far more serious, more somber story that dealt with real world issues. The rushed this thing into production, where it hit more behind the scenes issues than The Room (I'll save that for another time, otherwise I'd be here forever). In the end, new cast gets pushed aside while the old returns to the spotlight, the old cast never have to face the consequences of their actions (fans often accuse them of ignoring the consequences. Kira resorts to terrorism while the show tries to paint him as this hero who fights for peace) and towards the end the writing would derail itself with asspulls to justify them. Kenichi Suzumura, who played Shinn Asuka, has said in interviews that Shinn originally wasn't supposed to end up the bad guy, hence why he likes the versions of the character in Super Robot Wars games. It tried to paint itself as just as black and white as SEED's last act despite all this. Hell, I'd argue the ending is arguing against Tomino's work.
And in the end, despite high ratings and DVD sales it ended up killing the Cosmic Era (probably due to poor model kit sales. Those long-term fans were not amused). They went into the show wanting to cement it as the successor to the Universal Century, only for them to want to wrap it up with a movie that turned into vaporware.
As for other reasons, the character designs are mostly all the same. Everyone has the same face except for eye color and hair. The stock footage and flashbacks are a joke though. They were reusing SEED animation for Destiny and the fights generally devolve into mindless beam spam (the animators were very pissed at this). A lot of the characters lack humanity, feeling like stock anime characters rather than real people. And it has aged horribly.
Probably the thing that made the Cosmic Era so reviled in the West was the anime fans who liked the show. There's nothing wrong with liking it, I don't give people shit for that, but a lot of the time they would defend it with "well, the Japanese fans like it" and cite it's popularity (which I will take issue with). But SEED brought in it's own fanbase while alienating the older fans in Japan. We weren't seeing unsold piles of models and not hearing the cast criticizing the show. We don't get the fixed versions present in a lot of games over there, and any interview with Fukuda/Morosawa will have the translation and authenticity called into question.
If I were to judge SEED, it would get a C. Not great, but acceptable. It certainly doesn't live up to it's full potential or hype. If I were to grade Destiny, I would be sending it home with a note telling it's mother to meet with me.