There are multiple examples. It is difficult but apparently not so difficult that nobody attempts it. The accepted term for texts like that is "lipogram"; it's a popular form of the broader category of constrained writing.

Wikipedia has a page for "logology" which is apparently a term for the general activity of playing with language in that kind of way on a per-letter basis (so, including anagrams and palindromes, etc.)

https://en.m.wikipedia.org/wiki/Logology

If you sort all English words by frequency, the ones that contain 'e' would be distributed more or less randomly, so it seems more doable. Banning the first 1000 would prevent you from using the most useful words. I don't think you could construct sentences that wouldn't sound totally weird.

https://en.wikipedia.org/wiki/A_Void (from the original _La Disparition_)

is a noir/fantasy story, its actual plot is the story's lack of 'e'.

It's amazing. It's difficult and confusing to follow the writing, but it's an amazing book.

I know, it's fascinating. But it doesn't detract from my point.