I don't think I'd be alone if I said that I was a Marvel Zombie until I discovered Superboy and the Legion of Super-Heroes. This was my "entry" book into the entire DC multiverse.
1. The Legionnaires were the best and brightest of their era and their battles had HUGE cosmic villains and stakes.
Calling the Legionnaires just another "teen team" is severely underestimating them. "Teen teams" tend to be inferior versions of adult heroes; Teen Titans was only really interesting when they stopped being a "Junior Justice League." The Legion on the other hand, was IT: they had prestige and gravitas. They had A-1 clearance to any spaceport in the universe. The Key to the defense fleet of the UP was located inside Legion Headquarters. If the Legion ever actually was a future version of the Simon/Kirby "kid gang" as some have described it, they dropped it very, very early on in the Ed Hamilton years, when they saved Metropolis's first fusion powersphere, and Lightning Lad died fighting Zaryan the Conqueror.
In terms of physical power and sheer superior numbers, the Legion may actually be the most powerful superteam ever.
As a result of this, they fought terrifying cosmic-level enemies with truly terrifying powers: Mordru was Thanos, before Thanos was Thanos. All the huge cosmic Marvel stories of the 1970s (Kree-Skrull War, the Thanos War, Celestial Madonna) owe a debt to the Legion of the 1960s. This entire style of storytelling goes back to the fact that the writers had to come up with challenges big enough for the Legion.
Don't forget the Sun-Eater, which was going to eat the sun and cost millions their lives, or Computo building hundreds of computer versions of himself, or the introduction of the Controllers, or the first Khund and Dark Circle invasions of Earth. Paul Levitz took the Legion "big, epic story" formula and ran with it: there was the colossal Earthwar epic of his, and he dedicated the first few issues of his baxter paper series to a gigantic Legion of Super-Villains run by Nemesis Kid.
Not to disparage in any way a great team like the Teen Titans, but...how truly picayune and minor their stories were. When the Titans were fighting the generation gap, the Legionnaires were plotting their escape from the Stalag of Space.
2. The Legionnaires were "disposable."
This was, by the way, Paul Levitz's wording, not mine, so go get mad at him and not me. :)
Who was the knucklehead that called the Legion "the Archies in space?" Because the Legion has a strong undercurrent of death. Their stories were so high-stakes that it was entirely likely
Now in the Silver Age - the SILVER AGE, mind you, when death was much less common - the Legion had four deaths: Lightning Lad, the first Proty, one of Triplicate Girl's bodies, and Ferro Lad. There was also Kid Psycho, who lost a year of his life when he used his powers.
If you're a DC reader in the Silver Age, you're used to fakeouts and dishonest covers that screw you. But...when ADVENTURE COMICS had Triplicate Girl grewsomely vaporized by a tentacle of Computo - from the waist-out, WRATH OF KHAN-style...it wasn't a fakeout. Triplicate Girl was D-E-A-D, and stayed so.
Come the Bronze Age, we have Cary Bates's ERG-1 story where he dies at the end, Chemical King dying to prevent World War VII, Nemesis Kid butchering the hell out of Karate Kid, and so on, but I think you get the point here: the Legion was as a friend of mine put it, "a lot like the French Foreign Legion: you join for the exotic adventure, but you're more likely to end up massacred."
It isn't so much the Legionnaires could die, but WHY they could die: the Legionnaires lived four-dimensional lives. They could get married, they could retire, they could GROW UP with the reader. Compare how different this is, to...say, Superman or Batman: part of their problem is they have to always be young and unmarried so their pictures can sell beach towels or something, so no change to their status quo is entirely lasting.
Because the Legionnaires developed over time, and changed, and grew up, it was possible to really have an emotional investment in their lives. It was possible to CARE about them, because of how REAL they felt.
(This by the way, used to also be true of the X-Men, until the resurrection of Marvel Girl in X-FACTOR - the single greatest break of trust between creators and the audience ever performed.)