It won't hurt really nobody except the bumper spammers (and it won't need much coding)
Someone who has a really good reason to delete a post, can ask for it in the Meta (or report itself to a moderator)...
I use the edit button a lot (for legit reasons). Especially when I open a thread by myself, I want to be able to update the first post (and that can be months later).
I'd be okay to disable this feature for the lower ranks. If Copper members are excepted, I'd be okay to disable it for all ranks, and pay for Copper membership. But I don't think taking away functionality from legit users is a solution.