As the story goes, the Legion was inspired by Superman/Superboy and the other current-day superheroes (i.e., the JLA). But, you know what? I think it's the other way around.
Writers and readers often refer to the JSA in the Golden Age as a model for the creation of the JLA in the Silver Age. But, regardless of what was intended, it isn't. In fact, the JLA is in some ways the very opposite of the JSA.
The Justice Society of America (JSA) wasn't a collocation of DC's "big guns" like the JLA is (except for, you know, Geo-Force); quite the contrary. By definition, the JSA was composed of characters who couldn't hold their own titles. It was a testing ground for potential characters; the ones who seemed to be most popular might get a shot at their own series. And, when they did, they left the JSA (as Jay Garrick did). That's why "honorary" members (Superman, Batman, and Wonder Woman) almost never appeared and when they did took no part in the action; they would overshadow the real members of the JSA.
If you've actually read Golden Age JSA stories, you know that, unlike the JLA,the JSA almost never acted as a team, per se. They were, well, a society. Almost invariably, they were presented with a common problem at one of their meetings, then they split up, each member dealing with a different aspect of the problem. They coordinated their activities with one another, but, unlike the JLA, they weren't really a unified fighting force. Usually, they just got together at the end of their separate adventures to compare notes and laugh at what an idiot Johnny Thunder was.
You know what bridged the gap between the JSA and the JLA?
Yup. The Legion.
I added that last one in there for those few benighted folk who think, "the Legion is kind of like the X-Men." No; it isn't. The X-Men are kind of like the Legion. Marvel didn't invent groups of snotty mono-powered teenagers with complex relationships being nasty to one another; DC did. And, if you subtract the superpowers, Archie Comics did.Like the JSA, the LSH was/is composed of heroes who don't stand on their own (as in, have their own titles or solo adventures apart from their membership in the organization). Like the JSA, most of the members are pretty much "one-trick ponies", as opposed to the JLAers, who each have a variety of abilities or ways of applying their abilities (except for, you know, Geo-Force). Like the JSA, they basically got together because it seemed like it would be fun or interesting to work together; they are, at heart, a "club", just as the JSA was a "society". So, in those senses, the Legion is more like the JSA than the JLA is.
But then there are aspects of the LSH that presaged the way JLA works. Like JLAers, Legionnaires sometimes work on solo missions, but more commonly work in small groups with other members. Like the JLA, when occasions warrant, they fight en masse, and are experienced in teamwork, coordinating their attacks, using the powers in tandem, and covering one another's back. The Legion, like the League, is paramilitary.
In the Golden Age, the JSAers all had pretty much the same personality, just different powers. Yes, even the Spectre. The Legionnaires, over time, developed personalities (in the broadest sense) of their own: Brainy was impatient, Saturn Girl was a bit frosty, Lightning Lad was a jerk, Colossal Boy was none too bright, Bouncing Boy was upbeat, etc. While the JLAers started out like the JSA, stamped with the same personality, they followed the Legion's pattern of evolving the members as individuals.
Even more, the Legion pioneered the idea of particular relationships between individual members, i.e., some members got along with each other better than others. This is a pattern that JLA would also follow (e.g., Hawkman and the Atom were pals; Hawkman and Green Arrow were not). The JSA was never like that; they never disagreed about anything. Their toughest decisions were about who would go tackle dissatisfaction among the chewing gum workers at the Cleveland factory ("That Cleveland job's my baby!") and who would deal with fifth columnists infiltrating the paper clip consortium in Kansas ("It's Kansas for me, boy!").
So, although the Legion says they take inspiration from the JLA, from my perspective, it's the other way around! It was the Legion that paved the conceptual path for the JLA to be so different from the JSA, and, as the guardian of that transition, it has elements common to both groups.
P.S. Another note on JSA and the Legion.
How many times have I heard:
"Oh, Legion is so confusing and has too many characters!"
Okay, you got me there. If you have trouble remembering who Bouncing Boy, Lightning Lad, Shrinking Violet, Chameleon, Invisible Kid, Element Lad, Colossal Boy, Shadow Lass, Triplicate Girl, and Brainiac 5 are and what they can do, then, yes, you probably lack the attention span required to read a 22 page comic book. For Mort's sake, people! For most of the Legionnaires you just need to hear the names; you don't even need to see them to get the essence of what you need to know.
If you'll check Wikipedia, you'll see 37 Legionnaires, past and present (well, it's all future, but you know what I mean). If you do the same thing for the JSA, guess how many members you'll find: 37.
And the JSA doesn't have codenames that are nearly as mnemonic (Boxing Man! Cat Person and Bird Woman! Lightning-Fast Oldster! Medicine Man! Exploding Kid! Dream Man! Know-It-All 2!)
The Legion gets a "bad rep" because it's one comic into which an entire universe gets stuffed. Imagine that the entire present day DCU had to be represented in one comic book; it would seem pretty complicated!