Spell check was only a quick look-see for the obvious. It was my first scan. It was not my last.
My favorite go-to tricks to spot errors were 1) going through the manuscript backward, word by word, and 2) reading it aloud.
The going backwards part is the toughest, because it's slow going, plus you need to know homonyms/homophones, inside out. I wouldn't stop to read forward again for the sound-alike words--I'd only mark them with a red circle, to come back to on a second read-through, usually my read aloud run. But I always noted them.
Still, the going backward part "forces" your brain to look for spelling errors, and to see things like words that are in a sentence multiple times, which are typical signs of a revision/cut and paste error. As an example, If you see "words some typed phrase a typed," ("typed a phrase typed some words" when read forward) then you know someone reworded something or cut and paste and failed to get rid of a previous wording.
Reading aloud is also slow going, but it can catch similar errors that the backwards scan couldn't catch (like your missing word example).
Its more useful function is finding outright awful wording. Some writers can't "hear" dialogue in their heads the way people say it. By reading things aloud, you get rid of wording that makes no sense, or that is too clumsy and clunky. You want all of a manuscript, not merely the dialogue, to have the "flow" found in speaking the language itself. Some things just "sound right" to our ear (and our mental ear, too), while other phrasing with the same meaning never does.
So those were my "simple tricks" to proofread a doc. Worked like a charm for me, but YMMV.
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