A common literary way of trying to make an irrelevant character relevant, is to have them sacrifice themselves, to try to make a big impact and remove them from the story. It's not like the bridge couldn't be formed by the adventures of Link and Midna, trying to save both worlds. You could have completely removed Zelda from the story and not much would have changed, because the story largely focused on Midna and Illya. Your first meeting with Zelda, was basically a quest that was forced upon you in the exact same manner as OoT Zelda. After that she really didn't do much of anything that would insinuate she was her own character and not solely driven by plot convenience.
OoT Zelda was the one who sparked the whole disaster in the first place, trying to play hero, and anytime she was away from Link's point of view, she was working in the background influencing things and doing her own thing. You didn't always necessarily see her do it, but she did influence quite a lot of things from rescuing Ruto, to pointing Link in the right direction. The finale had her using her powers to help seal Gannon away and return Link to his own time.
Similar story in Wind Waker and Skyward Sword. Zelda was a spunky pirate that showed Link the ropes and when she wasn't directly helping Link, she was off doing pirate things and working toward her own goals. SS Zelda was the entire foundation of SS and everything Link did pretty much centered on her. When she wasn't busy scolding Groose and gang and helping Link out with his problems, she was off doing her priestess stuff completely independent of Link.All of the Zelda's sans TP Zelda, were so integral to the story that if you ripped Zelda out, the game wouldn't be able to function.
In TP, Midna and Zant pretty much influenced everything, with Zant causing the problems and Midna being the general guide, and Illya being the overall motivation to fight. Zant, Midna and Illya provided everything needed to advance the plot and inspire motivation to go through the great undertaking of becoming a hero and save both worlds. Zelda was bad enough, but tacking on Gannon was the real travesty and completely invalidated everything Zant did and making both him and Ganon feel completely pointless.
I absolutely do feel, given the amount of focus on everyone not Ganon and Zelda, that they were trying to step away from the traditional characters and do their own thing. No other Zelda game had that much emphasis put on characters that weren't Zelda, Impa, Gannon, etc. Except Majoras Mask. Illya and co. as well as Zant and Midna were fairly unique in that regard, in terms of having huge relevance in the overall story. The other Zelda games had new characters, but still largely retained the original cast, while TP threw almost all of them out and replaced them with its own.
As far as "dark and serious" MM was the darkest Zelda and handled tragedy beautifully. The entire game was basically one giant metaphor for "letting go" in terms of dealing with sorrow. Either you were bringing closure to individual tragedies or were fighting to prevent them, and not everything always ended perfectly and sometimes had bittersweet endings. Even the wonderfully designed time system, still drove home that you can't save everyone and weren't a miracle worker. The game was pure art and everything was very well done and in a way that wasn't overly indulgent or dreary and even if you don't like tragedy, you felt satisfied.
TP on the other hand, had a very sullen, whiny, and dreary feel to the overall game, that felt very reminiscent of what a teenage girl might think a tragedy is supposed to be. I get that a somewhat apocalyptic world, doesn't exactly inspire smiles, but MM literally had a moon hanging in the sky, reminding everyone they were going to die in three days and people still found a way to have some form of happiness and joy. A "dark and serious" thing, doesn't mean you have to be moody all the time and doing so, makes it less tragic and more grating. Nobody, even in such a world, can be miserable all the time and the idea that injecting some lightheartedness into something, makes it not serious anymore was always a really silly idea. Skyward Sword may have been the most lighthearted Zelda, but that actually made the sad stuff more poignant, when contrasted to the overall happy nature of the game.
My criticisms of Zelda applying to the other characters, was one of my major problems of the game and why I didn't like it in the first place. One of my favorite things of the franchise and part of what draws me to it, was all the interesting, fun, and lovable characters in them. Having most of the cast be sullen and whiny, made getting through TP a real slog at times. I just wanted to finish Midna's story and be done with the game and things always soured when Illya and co. showed up. Colin and Prince Ralis might be my least favorite Zelda characters of all time, by being annoying and completely void of any sort appeal or point. Midna, Zant, Agitha and the fishing lady, were the only bright spots I can currently think of, and as such, felt kind of jarring when juxtaposed against the pity party that was the rest of the cast.