I'm assuming being able to delete profile posts you have created on your own profile does exists as a permission, and my reading of it isn't incorrect (?)
The final value for the 'Delete profile posts by self' permission is 'Yes'.
Does the 'Manage profile posts on own profile' permission also need to be granted, perhaps?
There are no template changes that I believe would affect this.
None of the functionality of my add-ons leads me to believe they are the cause.