I think that worked because it didn't consume the show. The Dominion existed, but not every episode was centered around that war. The show remained pretty much episodic, with reminders of "hey, we have this going on", without being forefront every episode. Once in a while I rewatch an episode, and I am just blown away by the fact the "war" with the Dominion started so early in the show's run. It was a major plot point yet so subtle at least I could forget until it became prominent in the last season.

I think DS9 could have worked even in more serialized form--and, as you say, latterly it was pretty serialized. I'm not arguing that episodes good, serialized bad. I'd maybe come closer to arguing the opposite. I'm mostly agreeing that the mostly episodic TNG probably wouldn't get made today.