That's part of the problem, I think.
Everyone knows about the Kree-Skrull War, Coming of Galactus, Days of Future Past, Dark Phoenix, Age of Apocalypse, Infinity Gauntlet, the Winter Solider, House of M, Civil War and Spider-Verse. On the DC side, the most famous events are the Judas Contract, Terror of Trigon, Crisis on Infinite Earths and Throne of Atlantis. Only one of the above is specifically a League story, Throne of Atlantis. It's been adapted for the DCAU, Aquaman movie and in the JSA WW2 animated movie.
The Justice League has it's version of Civil War, it's called Kingdom Come. But that's AU and it doesn't get much exposure or reference nowadays. Everyone knows about Batman's plans to counter the JL being stolen and used by villains. But I doubt many people know the story is called JLA Tower of Babel. Most people know it from the animated movie, JL Doom. Out of the entire New 52 run with Johns. With him coordinating 3 different JLs for Trinity War. The only stories that made it in to a wider audience were the new origin (adapted as JL War, which looks just like MCU Avengers 1), Throne of Atlantis (adapted 3 times in 7 years), and Apokolips War (adapted loosely as Darkseid War DCAU). Meanwhile, Marvel Now (starting around the same time as the New 52) had Hickman's run on Avengers, New Avengers, Infinity and Secret Wars was multifaceted and acclaimed all the way through.
The only events I could find out of the Rebirth era, were JL vs SS and Doomsday Clock. Both of which are remembered for being unmemorable. After that, it's Scott Snyder's Metal and Death Metal. Neither of which were my cup of tea or where I think DC needed to be.
Going back to New 52 for a moment, the JL has a superior origin to the standard alien invasion story. It's called The New Frontier, but once again. That is AU and has no follow-up.
Let me be clear here. I'm not saying DC needs to adopt a present day Marvel style of event focused storytelling. Events like Original Sin, Fear Itself, Axis, Secret Empire, Siege (2010), Age of Ultron, Avengers vs X-Men, Inhumans vs X-Men etc. These events are nobodies favorites stories and do more harm than good, to the books that have to detour into every year's line wide crossovers. If the JL told good stories with interesting concepts, that could reach the iconic status of their contemporaries and past iterations. We could have something grand.
I also want to echo what @John Venus, said. The League needs their own villains exclusively and not just fight their rivals from their solo titles. That would force the writing to be creative and dump tired tropes.