I think it's valid up to a point.
But the playing while injured metaphor has some parallels as well. Not only is this a badge of honor, but it's flatly expected of you once you get past a certain level of competition. This is generally good and right.
However, past a certain level of injury, you're hurting both yourself and the team, and you're better off taking time to heal.
My personal experience with sleep deprivation is that if I go too long and too hard, I am doing less work than if I had rested, even in absolute (and not per hour) terms. And DHH is 100% correct that as most people get tired, their solutions to problems get worse -- that's definitely true for me.
I've tried various levels of sleep over the years, from a solid 8 hours a night to so little sleep that I would literally hallucinate during the day. For me, except in short (2-3 day) bursts, at least 6-7 hours of sleep a night is most effective for my ongoing productivity. Any less than that, and I would have gotten more done net if I had slept.
Having said that, everyone is different, and if you're one of those people who can get 2 hours of sleep on an ongoing basis and still be highly productive, then cool.