They can, and occasionally do. It tends to be people with a lot of spare time, or interested in memory techniques. And it probably helps if you know (ancient?) Greek, so it actually still rhymes. But it isn't super useful, so people generally don't.
If you paid people a living wage to recite the Iliad, you might get some more takers. We kinda do that, except it's actors in theatre plays or films. Sure, they don't take days to recite, but e.g. people put on several, different plays a year.
> It is certainly faster to look something up than to learn it
If it's faster to look it up, why bother learning it? Conversely, if someone had to look it up often, I bet it would be learned. Spaced repetition and all.
> The same with auto completion. Sure you can type faster, but it also means you, as a human, become less capable because you've never needed to hone the skills necessary to program without it. Programming without autocompletion is available in any text editor, even on paper.
I loathe this argument. Compilers or interpreters (computers in general) are sticklers for the exact spelling. Humans aren't; I still understood what you meant when you misspelled "Illiad". Why exactly spend the time and brainpower knowing exactly what something's called, as long as you have a rough idea of how the code works. Or, why spend the time typing it fully if you can have it auto-complete? When do you have to program in any text editor? Or even on paper, where the exact syntax can't possibly matter? (Equally, I'm sure that if this was a necessary skill, people could pick it up - occasionally they do, like for whiteboard interviews.)
Humans are very capable, and also lazy (one of the best engineering qualities). Don't mistake having no reason to do something with not being able to do something.