10K-ers, moderators and OP's can see deleted posts so they are available for moderation purposes.
There are a few reasons posts get deleted, either by clicking the delete link or because the Roomba got it. Red flagged posts also get deleted and hidden (thanks Catija) from direct view.
Users > 10K rep have access to the moderation tools and one of the lists there is recently deleted posts. This allows trusted users to audit the deletion of content and react to removal of anything of value, either by editing it into shape, undelete vote or discuss removal on meta.
Most posts that get deleted are not worth recovering but we can't rule out upfront that some might overlook a good post or some might even abuse it (delete your homework question for example after receiving an answer).
(almost) Nothing on the SE network is permanent deleted. Once you've posted content you gave SE also a license to publish your work. It does that by keeping everything you post in their databases. Being deleted is just another state a post can be in, much like closed. The difference being that deleted posts are only shown to those who are trusted to salvage a missed pearl.
I'm not sure if hide would be a better kind of wording then delete does now. I don't think it would make a difference in how easy or difficult the actual feature can be understood by new or seasoned users.
As for the grayed out option: we already do that for seriously down voted answers.