妨げる and 阻む seem to be synonyms (to prevent or blocking somebody for doing something). Indeed, many dictionaries have this idea of using both verbs in their definitions:
阻む:
進もうとするのをさまたげる。
妨げる has an extra definition i think: “To disturb someone in a bad way, not just blocking an act or an action, but being troublesome or annoying”
遮る perhaps is the most neutral word among all. Just avoiding something to happen, to not let something occur, but without intentionality. Isn’t this the idea of 阻む?
I feel like the most common situation I see this word used is in novels when person A is speaking and person B interrupts them – the verb used to describe what B does is 遮る. (The dictionary says it can also be used when something’s blocking or obstructing a view or the wind, and there’s certainly plenty of examples of that usage too; either it doesn’t come up as often in what I read or maybe it just doesn’t stick in my mind as much…)
…and having said that, look what appeared in the book I’m reading:
雪路はまぶしさに思わず手を上げて光を遮った。
Funny how words you could have sworn you rarely see suddenly become salient when you’ve got them on your mind.