If you're using git-shell or git-daemon to host your git repositories on a server, then you can't use that alone to also manage transfer/access of stuff that is managed through git lfs.
But beyond that, what’s keeping the server from storing these deltas instead of full versions? Only argument I can think of is that server side corruption of a single file would break the chain of deltas and you’d lose every child.
But surely that can be solved using some sort of redundancy.
I'm making a simple .zip over HTTP solution for my MMO now.
It just has a naming scheme with dates and you send your last .zip date to the server and the server sends a list for you to download.
The good part with dates is that you can consolidate the data retroactively without breaking the system!