This seems the most likely to me, purely based off the fact that going on without sleep gives the same feeling in the head as walking a long time gives me in the foot and legs.

So, the brain is an electro-chemical engine, and like other engines, it must be shut down to perform regular maintenance so the engine lasts longer, and is less prone to error conditions.

Not just the brain, but the whole body.

If I go without enough sleep, my hair and skin start getting greasy, my immune system is weakened, my muscles start feeling tired, etc.

Your hair and skin get greasy because you aren't bathing, not because you aren't sleeping.

You are incorrect, as you can trivially verify yourself with a few days’ first-hand observation.

Even if you don’t bathe at all for a few days, or just rinse yourself with water (skip the oil-stripping soap for this experiment), the greasiness of hair/skin fluctuates noticeably throughout the day, and increases substantially after a day or two of insufficient sleep. (That’s right, the greasiness will go up and then subsequently go down, without your doing anything special in between.)

[Side note: it’s pretty disappointing when people make confident sounding assertions about things they clearly don’t know anything about and didn’t bother to do the tiniest trivial bit of study. I find this happens quite commonly even in discussion venues where there’s ostensibly a cultural value placed on reasoning and evidence.]

I've been sleep deprived for extended periods before and never noticed the difference. Is there sample size > 1 evidence for this you're aware of?

You want to casually study myself while sleep-deprived? That sounds like a recipe for trustworthy results, I'm sure.

Of course I'm going to get more greasy after two days of not using soap. I don't see how that would verify anything that you are claiming.

It functions as a regularizer in learning and reduces unnecessary energy use.

While those are benefits of sleep, I don't believe we consider them to be the reason we sleep.