I don't think there's any evidence of proposed theory of a benefit of missing sleep. However there is something to be said for sleeping according to natural cues rather than culture and there may not be much point trying to force yourself to sleep. Part of the problem is that we typically don't have the freedom to sleep whenever we choose so any missed sleep rarely gets caught up and we move towards chronic lack of sleep.

My belief, such as my opinion is, is that our true strength is in our adaptability. We should be able to cope without sleep on occasion, and if we allow ourselves to go the other way sometimes as well we allow our bodies to find their own balance amidst a chaotic environment. Much the same philosophical approach as I take with eating, though the result is quite different. There's science to suggest that periods of fasting are used physiologically, and I'm not aware of anything supporting lack of sleep in the same way. However evolution is not about perfection, it's about survival. Fasting was enforced by food availability. If there was a benefit to survival it would be a side-effect that would grow in the population. Lack of sleep would have been necessary sometimes, and I believe we would have evolved to be able to cope without it being too detrimental to our health, but that doesn't mean we should reenact it. The main advantage would have been in being able to survive the immediate danger simply by being awake. In our modern lives, we can gain short-term advantages by being awake for more hours of the day. But we already push this to the limit (or beyond) anyway. Ideally, we should want to be in a position where sleep (or at least rest) comes to us easily every night. We can manufacture enough random 'stresses' to keep ourselves sharp.