My idea would to split them into two generations. Generation one would be the World War 2 generation with Alan Scott, Ted Knight, Kent Nelson Dr. Fate, Charles McNider Dr Mid-nite, Johnny Thunder, Hawkman & Hawkgirl, Ma Hunkel Red Tornadoe and basically all the Golden Age characters that fit in the era they were created in alongside all the original Freedom Fighters, Seven Soldiers and all of them collectively known as the Squardron Supreme. Just like the comics, the team disbands in the mid-50's due to the communist witch hunt.
Then there would be the second generation which was active 20 years before Superman debuted and operated for 10 years before disbanding. The new JSA would include Jay Garrick, Ted Grant, Dinah- all the guys whom you want to believably be old but still active in the present day alongside legacy heroes like Rick Tyler and Jesse Chambers who could be nebulously be described as the descendants of the originals. Hector Hall has enough weird stuff going on that his longevity could easily be explained through magic. Lyta's origin could be simplified, remove her blood ties to Diana and have her be an exiled Amazon whose heritage is kept a secret from the world. Pieter Cross or Beth Chapel could be the new Dr. Mid-Nite, thankfully they don't share a blood connection with the original. Michael Holt could be a part of this generation. as the tech guy. Connecting the two generations together would be Alan Scott who has the best explanation for his longevity and his inactive period between the first and second generation could be explained in a number of ways from losing his powers not his immortality or trapped in another dimension.
Then you could debut the third generation with Power Girl, Mr. Terrific (Michael Holt), Dinah Laurel Lance. Dr. Mid-night (Beth Chapel), Jade, Obsidian, Atom Smasher, Star-girl (Courtney), Wild cat (Yolanda), the android Hourman, Rick & Jesse's kid, Hector & Lyta's kid and any other character you would like to see.