I think others have mentioned TrueNAS as an OS for your Desktop. TrueNAS uses ZFS which is nice cuz it uses RAM to help with speeding up file operations. TrueNAS makes it easy to set up NFS shares if your laptop is running a Unix-like OS, or Samba, for anything else. IDK how much RAM is in your desktop, but if you can get that to 16 or 32 GB, you'd be set on that front.
For dealing with an OS you don't want to change, I'd think about the following first:
What is the speed of your network card in the Desktop? If its 100mbit, you may want to look into upgrading that, if possible to at least a Gigabit card for PCIe. That would speed up anything you do with it.
As for needing local sync for when you're away from your home network, Syncthing could maybe do what you want. TrueNAS can run Syncthing pretty easily as well, but it can be installed on anything, though, idk how this works if you set Syncthing to track a folder thats also one mounted on your local machine via NFS/Samba. Syncthing will just sync the most recent changes to a file to the server, so you can sync when you're on your home network. Assuming that no one will modify the same files on your desktop when you're away.