I think that the long rest rules are worded that way because you don't have to spend the entire long rest sleeping to get the benefits, and that there are times where you can take a long rest without sleeping and still get the benefits. And no, it doesn't explicitly say that you eventually have to sleep. If they said you had to get 'x' hours of sleep per long rest, per day, or every 'y' days, it would 1) unnecessarily complicate the rules, 2) set a specific rule when even in the real world these specifics are variable, and 3) add little to the game. Most of the time a long rest can be glossed over since nothing happens.
But in a question about whether elves need only a 4 hour long rest instead of 8, Mike Mearls responded: "nope, they still need 8 hours of rest, but spend only 4 of it zonked out" implying that non-elves are zonked out for at least that and really longer based on the trance rule.
I certainly would rule that after the first day of no sleep that there's a chance, if not a guarantee of suffering exhaustion. And it would not be restored at the end of the long rest that caused it. Because the cause would be not resting during the long rest. In order to determine that, the penalty would be applied after the rest (and restoration from it) was completed. I hadn't gotten into the sleeping in armor thing yet, but the idea that you can't regain from exhaustion sounds like a good rule too.
As for concentration, I have no problem allowing a caster to maintain concentration during a short rest. While sleeping (or an elf in reverie), no. And I don't see a problem with that, you decide whether you risk exhaustion to maintain a spell or rest. Hex lasts up to 24 hours, so starting the long rest a little later isn't a big deal. But it all should play to the story.
Randy