I am such an idiot. I can't believe Brad fooled me like this! And please forgive me if you figured this out already yourself.
But I didn't figure it out until this morning.
The Legionnaires aren't in the present to resurrect Lightning Lad.
They are here to someone else entirely, using lightning.
Someone who helps New Earth discover Earth One (the Bronze Age version of Earth); someone who will in the near future make it possible for New Earth and Earth One to team up to (somehow) preserve the version of the 31st Century that these Legionnaires call home. That's why they had to come to the past.
In fact, Lightning Lad is surely the final member of their team and should arrive any issue now; he's probably the source of the lightning they use in their resurrection scheme, and I wouldn't be at all surprised if he loses his arm in the process.
It's the only thing answer that puts all the puzzle pieces into place, including why retailers have been encouraged to double their orders on an upcoming issue of a particular comic.
Nice one, Brad; very, very nice. I owe you one.