Matthew,
Re your questions below:
What do you feel makes/made these books so popular?
Are the mysteries really so magnificent?
Cute characters?
Books came out at a time when entertaining people was easier?
Well, if by "mysteries" you mean "puzzles," then no, that's not a factor in their popularity at all. Because, if you read them, not only are the puzzle aspects
not all that magnificent, but comparatively few of them are puzzles at all. They're action/adventure chases, with the identity of the villain the two sibling sleuths are pursuing known early on. My understanding is that the stories featuring the Hardys' distaff counterpart, Nancy Drew, are more likely to be whodunits, but, like most boys, I didn't read Nancy Drew growing up.
Nor can I call the characters "cute," precisely. But they are intriguing to a young reader. Always involved in danger and adventure. Their father having a glamorous profession. Both their parents trusting them implicitly, their father to assist him in his business, and their mother to be able to survive any dangerous situation that book's adventure may find them facing. They're just old enough (from the perspective of reader of 8 to 12, the main target audience), that it's not altogether unbelievable for them to be involved in all that mystery and intrigue, but still young enough that the readers can identify with them. They had their own car, their own motorcycles, their own boat, and they were experts at handling all of them. They were alike enough, and close enough in age, that they're believable as best friends, but different enough that they're believable as brothers. And the format is loose enough, that dozens of different "ghosts" have been able to contribute to the series under the "Franklin W. Dixon" house pseudonym.
How they've managed to hang on this long when all the other "Boys' Adventure Novels" that the Stratemeyer Syndicate put out, Tom Swift, Don Sturdy, the Boy Allies, the Motor Boys, etc., etc., etc., have long since passed from the scene, I can't really say. But the fact that they have hung on, and adapted to changing times and changing styles to remain immensely popular, suggests to me that it's not just a matter of nostalgia, of having been created "at a time when entertaining people was easier." That probably has something to do with it, of course, as parents who read the books when they were young are likely to pass the books on, or recommend the books to, their kids, but it can't be a major factor, or new books wouldn't continue to be published.
Two other points. The Hardy Boys, for some reason, seem to have a broader appeal than Nancy Drew. According to what I've heard, while girls will read the Hardys, boys almost never read Nancy Drew. That bears out my own experience. I read the Hardys avidly, but never cracked a Drew novel.
I'm already on record as naming
Dick Tracy as my particular crime fiction "gateway drug," back, in fact, before I could even read. But the first mystery novel I ever read was
The Wailing Siren Mystery, a Hardy Boys mystery that feature a cover illustration that had Joe Hardy duking it out with a gangster on a boat in the middle of Barmet Bay. So the Hardy were certainly there in my embryonic stage of becoming a mystery fan.
Second, the Hardys never struck me as particularly "cozy." In fact, given all the gunplay and fist fights, and dangerous gangsters, etc., they strike as, if anything, more hard-boiled than cozy. Their father's a private eye, for crying out loud, not a British aristocratic who solves crime as a hobby. I just mention this because I couldn't help wondering why you put this thread in the "cozy" section.
Finally, a hearty welcome to MWF.