I have a very similar setup to yours, a relatively large music library around 1.7TB of mostly flac files on my server. I'm able to organize these files locally from my laptop, which at various times has run either OSX, various GNU/Linuxes, or Windows. However I do not bother pushing the files themselves back and forth over the network.
Even if I did, I wouldn't automate the syncing, I'd only run it manually after I'd done my organizing with Picard for that day. After all, it the organization with Picard isn't automated, why should the syncing be? I'd probably use rsync for this.
In actual practice I do this: Connect to my server from my laptop using ssh, forwarding X. Run Picard on the actual server through this remote connection. Picard runs just fine over ssh. Opening a browser from a Picard tag for occasional Musicbrainz.org stuff is a little slower but works. I would then use a tmux or screen session to run the rsync command when I'm done with Picard for the day for syncing to a backup if necessary.
I don't really bother keeping a whole copy of my music collection locally on my laptop or phone though, since It's been bigger than is practical for a long time. Managing multiple libraries and keeping the two in sync turned into such a hassle that I was spending more time organizing than actually listening (or making mixtapes/playlists). To listen to my music locally I've used either Plex or Jellyfin, sometimes MPD (like when my server was directly connected to my stereo receiver), or just shared the folder via samba and NFS.
Yeah, that was a typo. Thanks, I'll fix it.