We've seen Johnny Thunder tell Wally West how he (Johnny) wished his Thunderbolt would get the JSA members away from the Senate HUAC hearing, and nobody remembers they existed since that point in time.
A weird idea for where they went (and how to get them back) would borrow from 1972's Justice League of America run by Len Wein, and the story from issues #100-102. In that year's annual JLA/JSA team-up, the assembled heroes went in search for the missing members of The Seven Soldiers of Victory.
What happened to this group who had also been known as The Law's Legionnaires?
Turns out the members of the group had found a way to defeat the Nebula-Man, but it resulted in the death of one of them. The remaining members were removed from the (then) present-day time and cast back to various points in the past, plus the world they had lived on quickly forgot their existence.
Now, I doubt DC would do this, but it might be cool if when Thunderbolt spirited the members of the JSA away from that hearing, Thunderbolt and the other JSA members were removed from time and memory by Dr. Manhattan, cast off to be stranded by Dr. Manhattan at various points in the past. (Jay Garrick's appearance in "The Button" crossover could then be a failed attempt by Jay to break free from wherever/whenever Dr. Manhattan had trapped him.)
Like I said, it was just a thought.