I think the thing here is (possibly) "define a few" (Ly). What's the difference in distance between Sol and the planet with the base on? A 'few' `Ly could be defined as 10, 50, 100... Loose definition I know, and gut instinct dictates to me that with no defined distance, it could be quite large compared to how far (and how long!) they'd travelled.
I'm not a native speaker, but "a few" usually means <10...
Also: check the 'goldilocks' zones for the star types. What if the Thargoids had nuked all earth-like worlds to prevent human colonisation? What if the earth-like-planet went bad by the time they reached there? Should be candidates to check. After all, it was many years ago. I am sure there are other possibilities.
Example: Humans see a planet capable of giving them an ELW. They take off and fly to it. It takes 700 years. However, 650 years before they arrive a city-sized comet enters the ELW atmosphere, a la Chicxulub.
You really think FD would give us such a nested riddle to solve? Including some highly unlikely scenarios? Naah. Impossible for me.
Personally I think the riddle is fairly simple and easy. There should be an ELW within distance of a few (<10) LY from the site and the generation ship would be left in orbit of the ELW. Else it's impossible for us to find.
As stated before, systems around weren't thoroughly checked. So that's the goal for us I presume.
And, more on the "a few" ~<10 LY: Assuming they travelled from Earth which is ~160 light years from the site, they missed their target by a few light years. So little they say. So it can't be another 50 or 100 LY. It just can't. They are almost at their destination. Almost - I'd say 5-10 LY, no more. Also, they were on their own, they fixed the ship, got the fuel somehow - they would not be able to make another long journey. The opposite - they travelled to a system next door at most... Or they just didn't make it and are still in the system, like Voyager probe. That would be the easiest version to solve and most plausible, since we can't get a good candidate for an ELW nearby...
Transmission even suggests that the ELW should be within the system - they could stare at it, and they did it during all those long years. I don't quite understand this part.