If you're interesting in talking about this a bit, I bet I can prove it to you! I've always heard of SWTOR (and to some extent The Secret World) as extreme exceptions to the rule when it comes to MMO storytelling. But it's not "MMO" exactly that makes the difference.
If you don't mind, what MMO plots would you consider good, and beyond that, what are the games you've played in general you consider to have the best stories?
Well, I'd like to preface my statements by saying that I'm a characters-first reader/player when it comes to story. I much prefer a tale rich in characters and light on plot complexity to the alternative. So Destiny essentially fails by my criteria before we even come to any consideration of the plot, which is actually somewhat interesting in places (Rasputin especially).
That said, I'd point to City of Heroes/Villains as another MMO in addition to the two you named that offers much more interesting story beats than Destiny. There wasn't much to start with, of course, but even in the beginning there were interesting characters and plot lines. They also later went back and totally redid all the starting content, offering players a ton of choices in what quest lines they'd like to follow, some of them quite good. But I suppose it's somewhat unfair to compare Destiny to the final product CoX became.
Speaking more generally, while most MMOs are known for an unobtrusive story, it's generally much more fleshed out than Destiny's. When you complete a quest, you'll generally get some sort of story-based acknowledgement of that fact, even if it's just a few lines of text. More impressive achievements will generally receive larger, more elaborate response, up to an including cutscenes. Destiny... doesn't really
do that. Consider The Dark Beyond. You go to the Moon (apparently quite an achievement at that point, since traffic has by and large ceased), you find a dead Guardian (who really should have been named, btw), fight off a massive wave of baddies, and return with his Ghost to tell the Speaker the Hive is planning an invasion. You get absolutely zero story acknowledgement of this. Not a single line of dialogue. For a story-based player, there is
no payoff.
No, as to the last point, favorite stories? Well, that's tougher

Halo is one of my favorites (which is why I'm so disappointed in Destiny), Mass Effect, Rise of Legends (which nobody but me remembers

), Dragon Age, City of Heroes (again, some of the later stuff was fantastic), KOTOR, and Jade Empire probably give a decent sample. I like colorful characters and flavorful worlds more than I like deep, meaning-laden plots, but Destiny doesn't really provide any of that.
Kind of jumping off of your statement, I look at Destiny as some sort of badly-executed intro to MMOs with a great implementation of a raid within a first person shooter.
However, I don't know who they're fooling thinking 3 raids is enough for 2 years.
There's probably going to be additional expansions after the first two.