Since hit points are explicitly not entirely physical, I don't see a problem: You're flat-out easier to kill if you're dragging your hind end. I could be sold on using Wisdom instead of Constitution, though.
IIRC, it's about 15 days before you die from lack of sleep. Based on that, I'd say a Wisdom save every day, with a DC of (15 + sleepless days + current fatigue level) would be appropriate for absolutely no sleep. You can grant a bonus on the save of +2 for each hour of sleep the character gets. Checks start once you either miss all sleep in a night or go three days with less than half your required sleep. Any night in which you get a full allotment of sleep does not require a save and resets the sleepless days count.
That system makes that 15 day mark something of an inevitability, if you can't get sleep. It also grants some pretty good bonuses for getting any rest. It's totally not play-tested, but I don't expect most parties to get beyond the first couple levels. It's just not that common. Instead, having rules encourages the players to manage something the characters would almost be forced to do.
I'd probably only bother checking for falling asleep during a short rest. Make a Wisdom save vs. whatever the current target is or lose the benefit of the short rest because you didn't actually get a chance to bandage wounds, or whatever. Additionally, you are automatically surprised should anything happen during the rest. Any other impact is easy enough to handle by the level 1 impact of disadvantage to ability checks (including surprise/perception and initiative) -- at DMO, the character might have to make more surprise checks than usual, just to be sure.
In the link Xeviat gave, a guy was mentioned to have gone 6 months without sleep before he died.