Can a PDF be embedded  in the posts?

technically probably yes BUT why would anyone want to?

pdf is one of the most annoying formats ever invented!

Companies that create public websites where most of the content is in pdf files not only end up making that content inaccessable to some phone users who might not be able to render pdf but they also make themselves look totally incompetant

I guess any such addon should probably should also be able to detect pdf attachments in posts and put them in some storage space so that a link can be shown to users to download them for printing on paper. (is there any other use case for pdf that might make sense?)

I don't think pdf should ever be embedded in html contexts where the user might be expected to have something to render it.
If posts like that start turning up here I'd probably want those to be treated the same way as above.

I would not want anything here to encourage users to form such bad habits!
Most browsers have native PDF rendering these days (in javascript, not using the buggy Adobe proprietary plugins). If yours does not, there's a link to download the file. From a systems and security viewpoint it's an order of magnitude less annoying than a YouTube or Twitter embed.
It happens to me too sometimes. But don't you see the alerts in the notice field ? They tend to be there despite of comments and likes being there in feeds or not.