I've noticed that when I am tired, my chess playing gets even worse than when I am not. I blunder more often and get upset at myself the next morning, ergo I do not play chess when I am tired too often.
Chess and sleep-deprivation

I've noticed that when I am tired, my chess playing gets even worse than when I am not. I blunder more often and get upset at myself the next morning, ergo I do not play chess when I am tired too often.
It's a gamble. There's different modes of exhaustion and I think different parts of your mind can be fatigued, and I suspect that if parts of the brain are fatigued that can cause trouble for you while other parts that are an asset to chess are not overly fatigued that it can actually help. But I also think that it is a difficult thing to manage and predict, and therefore a big risk. But if you play tired often enough (which I don't recommend doing), you may see what I mean. Obviously, the odds are in the house's favor, so though you might win sometimes, eventually you'll lose money (ELO).

1% is a small difference .....
....but you are being competitive that can mean the difference between winning and losing. If your 1% more fatiqued than your opponent or 1% under what your brain needs to think properly is huge here.
I have played in leagues and came directly from my job with tie and shirt on and etc. For sure they thought they had an easy win as they thought i would be fatiqued in the game. Yeah you may say that is not sleep deprivation ...... hmmmm. Well in either case you can prepare for it. Work out and it does not have to be weights just something that gets the juices flowing. And vary the speed of the workout during the workout. Works like a charm!

I'm really terrible when I'm tired. My brain then just refuses to analzye positions properly and I make all sorts of blunders.

I've noticed that when I am tired, my chess playing gets even worse than when I am not. I blunder more often and get upset at myself the next morning, ergo I do not play chess when I am tired too often.
It's a gamble. There's different modes of exhaustion and I think different parts of your mind can be fatigued, and I suspect that if parts of the brain are fatigued that can cause trouble for you while other parts that are an asset to chess are not overly fatigued that it can actually help. But I also think that it is a difficult thing to manage and predict, and therefore a big risk. But if you play tired often enough (which I don't recommend doing), you may see what I mean. Obviously, the odds are in the house's favor, so though you might win sometimes, eventually you'll lose money (ELO).
Upon further consideration of your original post, I know that a friend and I have gone to a couple of high school chess tournaments on the chess team where we roomed together the night before and spoke until one or two o'clock in the morning when the tournament started at eight. The difference is, though, that I always felt well-rested that morning. Coincidentally, I play better at those tournaments than I do at any other time. I always assumed that it was my feeling the pressure and being more careful, but I think that your thoughts of finding that "sweet spot" so to speak, could also be an alternative answer.
So I noticed that when I am tired, compared to a normal day, I make a lot more blunders and mistakes. Whereas I only make 1 or two mistakes wide awake, I work a morning job and am an insomniac. I took a study for it, and make about 5 mistakes and 3 blunders while tired. I sleep for an average 3 hours, work 7, and stay awake the other 14. Not the best of times
I've noticed that being fatigued can be a help or hinderance to the game. It is generally bad, but sometimes it is an asset. During blitz, late, when I am tired, I sometimes get into a quasi-soporific trance of focus and lucidity, wherein I'm not as impulsive or caught up in the game, and as long as I'm not too far gone, lethargic or in a stupor, and I can actually win a much higher % of games. Next morning - not so hot. Takes awhile to hit my stride.
My worst problem when playing chess is consistency fickleness of biological organism.