At my dayjob we used Git LFS for a bit, but foud it unworkably clunky - we eventually found it easier to just make a separate "LFS" repository and add it as a submodule to the main monorepo. Now we can rewrite the history of the LFS repo on an as-needed basis.
That's one of the features of Git LFS is separately managed storage and lifetime.
You are correct that GitHub does not offer that feature.