Here's the thing though. We already have a 'modern' post-WW2 JSA...its called the JLA!
The League was basically the Silver Age reboot of the Society (hell, the only reason they used the word League and not Society was because the latter sounded too old-fashioned ).
When the JSA were brought back on Earth 2, it was basically as an alternate universe JLA (with a side-helping of Golden Age nostalgia of course, which back then was like the equivalent of 90's nostalgia today). Roy Thomas of course emphasized their WW2 roots more deeply in the years before COIE, and Post-COIE, once they were in the same reality as the JLA, the legacy aspect of them being WW2-era forerunners to the JLA and other modern heroes was further cemented.
So that's pretty much where we are right now. The only things that make the JSA unique compared to the JLA are either that they are from another universe (which is hardly that unique in an era where the Multiverse concept has gone mainstream in pop-culture), or that they are from WW2. And another appeal of the WW2 aspect is that it allows DC to retain some of its actual Golden Age history. We can't have Siegal/Shuster Superman or Kane/Finger Batman in the 1930's, but we can at least say that Alan Scott and Jay Garrick were flying or running around back then, as published in the original books!
That said, adaptations like Smallville, and the CW's Stargirl (as far as I'm aware) have moved the JSA to more modern eras while keeping the legacy aspect intact up to a point. But those will always be seen as 'alternate' takes. The real JSA will always be perceived as the ones from WW2...that's just the pop-cultural niche they hold.
Yeah, its more or less in line with how I think about it. I'd just add 5 years on to every part of your timeline.
That said, after all this discussion, I decided to lay out my own take on the DC timeline. This is mostly a Batman-related timeline, but I've added in a few mentions of other characters/teams. Its by no means perfect (especially the stuff with non-Batman characters), and I'd love some feedback on it!
Here goes:
Year 1: Bruce debuts as Batman, Clark debuts as Superman
Year 2: Joker first appears followed by other rogue's in Gotham, Batman and Superman first meet
Year 3: Dick becomes Robin, Diana comes (back?) to Man's World as Wonder Woman, Oliver debuts as Green Arrow
Year 4: Classic Dynamic Duo era
Year 5: World's Finest team-ups with Superman and Batman (and Robin), Kathy is active as Batwoman
Year 6: Barry debuts as Flash, Hal debuts as Green Lantern, Arthur goes public as Aquaman, Justice League is formed
Year 7: Wally becomes Kid Flash, Robin meets the Justice League, Teen Titans are formed, Barry meets Jay (on another earth?)
Year 8: Barbara becomes Batgirl, Dick goes to college, Batman meets Talia and Ra's al Ghul, Damian is born
Year 9: Hard Traveling Heroes with Hal and Oliver, Death/Return of Superman
Year 10: New Teen Titans formed, Dick becomes Nightwing, Jason becomes Robin, Clark and Lois get married
Year 11: Crisis, Barry 'dies' and Wally becomes Flash, JSA returns to Earth 0(?), Jon Kent is born
Year 12: Barbara paralyzed and becomes Oracle, Jason is killed by the Joker
Year 13: Tim becomes Robin, Jean Paul-Valley temporarily takes over as Batman, Hal becomes Parallax and Kyle becomes GL
Year 14: 'Watchtower'-era JLA forms, Electric Superman
Year 15: Gotham becomes No Man's Land
Year 16: Jason returns as Red Hood, Hal returns as GL, Infinite Crisis
Year 17: Kate becomes Batwoman, Damian meets Bruce, Bruce is lost in time, Barry returns as Flash during Final Crisis
Year 18: Dick temporarily becomes Batman, Tim becomes Red Robin, Damian becomes Robin, Bruce returns and forms Batman Inc, Barry causes Flashpoint, Supergirl appears, Barbara returns as Batgirl
Year 19: Damian dies and later returns
Year 20: Bruce temporarily loses memories and Jim Gordon becomes Batman, Bruce later returns, Jon Kent learns who his father is
Year 21: Barry reunites with Wally and learns about the timeline changes, Jon becomes Superboy, aborted Bat/Cat wedding
Year 22: Jon returns from future aged-up, Alfred dies
Needless to say, the whole thing is a mess, and a bit constrained by my lack of knowledge about some events and stories. But it mostly holds together.
The biggest issue IMO is the circumstances of Jon's birth, which pushes stuff like the Death/Return of Superman, and the marriage into the Pre-Crisis era, and messes up the relative ages of Conner (as per his original origins) and Jon. And not to mention Hal becoming Parallax which by my timeline occurs years after the Cyborg-Superman showed up. Then again, I'm not sure how much of Superman's past continuity still applies verbatim. This is admittedly a more Batman-biased timeline.
And the other issue is when exactly do various other JLA members, apart from Superman and Batman, first appear. I think this can be done almost randomly within the first 5 years or so, depending on how old/young you want to make them.