I really don't think there is anything wrong with the format, merely that the writers of VOY and ENT were lazy and bad and most of the actors mediocre. Though I don't mind breaking the format a bit, as long as it doesn't mean breaking the certain premises that I find central to Trek. When I watched TNG, I never thought that, you know what, this show needs is more war, religion, capitalism and morally shady main characters, but apparently some people though just that because DS9 happened (and a lot of viewers though that too, as DS9 was somewhat popular.) Both VOY and ENT had interesting premises, but they did very little with them. ENT in particularly was disappointing, everything being new to the characters and technology being way more limited could have given fresh perspective to 'boldly going' but that rarely happened.
I think exploration and meeting aliens and weird phenomenons is central to Star Trek, and it is possible to write good stories around that (and if you can't then perhaps you shouldn't be making a Star Trek show...) I really hope this is what we see in the second season of Discovery.
Eh, I dunno. Compared to the cast of a modern drama, DS9's characters were still a bunch of saints. Quark pretended to be nothing more than a Gordon Gekko type character, but it became clear as time went on he had a moral compass. Garak did many terrible things, but he was ultimately a heroic character. No one else out of the core group of characters was really all that morally gray.
As for the Dominion War, the extent to which it took over the later series was partially the fault of the studio, from my understanding. They originally planned for it to be a one-season affair, but when Worf was reassigned to the show they had to push back their planned schedule, which ultimately resulted in the war not finishing up until the end of the show.
In general I feel like DS9 was the most "adult" of the Trek series because it found its strength in moral ambiguity. I'm not talking about the common modern trope in fiction of flawed cast who does bad things. I mean that although it kept to the classic Trek pattern of having "issue" episodes, it seldom ended the episodes with a moral sledgehammer of "X is right, and Y is wrong." Instead it openly explored all sides of an argument and showed there really are no pat answers. This is part of why I was so disappointed in some of the choices in the endgame of the series involving the Prophets and the Pah-Wraiths, which put a good/evil duality into the show - indeed in to Trek - that had never existed before.
I do think there are lots of exploration-related concepts which have not been really covered by Trek. That said, I do not think the current writing team is likely the best one if they want to go that route. If you want to have a writing team which can think of original and creative sci-fi concepts, you need to hire sci-fi writers, not people who have written episodes of Law and Order and Desperate Housewives.
Over the years, there are many many episodes of TNG I have rewatched, even some VOY (Year of Hell) and ENT (In a Mirror, Darkly), but not once did I feel the urge to rewatch any of DS9 episodes. To each their own I guess.
Those aren't even among my favorite for either series.
My favorite VOY episode - by a wide, wide margin - is Living Witness. Most of the other ones I liked were the "character focus" episodes (Barge of the Dead, Before and After, Mortal Coil, Someone to Watch Over Me, etc). Voyager's attempts to do action were always mediocre, IMHO. I could simply never suspend my disbelief and get involved in the idea the ship was actually in danger.
For ENT, I consider the "Vulcan trilogy" in the 4th season to be the high point of the series. IAMD is fun, but it's comic-book schock, and I like my sci-fi more cerebral than that.