This is unexpected behaviour from Github here which may (and has, by the anecdote of OP) cause permanent data loss. Documentation is not good enough, as users should not have been expected to have read the entire documentation.
On that note, an organization admin can _directly_ delete your private fork without even deleting the source repository if they want. GitHub's permission model is fairly direct that private forks you make through your membership in an organization are more the organization's property than the forker's.
This is exactly how it works today already. If I try to delete a private repository people have forked, I see the following:
> We will also delete all 4 forks since this is a private repository.
Clicking on the delete button, again:
> Unexpected bad things will happen if you don’t read this!
> This will also delete all 4 forks since this is a private repository.
> [type name of repository]
To this day I thought that the "fork" concept was only a relationship at the level of UI, but as I see it has a logic in it, that is the fork depends on the original repository even for permissions, and that to me is surprising!
If the org doesn't work this way, it can disable forking so that it's not allowed at all on the repo (or org-wide), like you said.
I know this isn't common but I actually use a unique user for my company "myname-company"