For local backups I use this command

$ rsync --update -ahr --no-i-r --info=progress2 /source /dest

You could first compress them, but since I have the space for the important stuff, this is the only command I need.

Recently I also made a migration similar to yours.

I've read jellyfin is hard to migrate, so I just reinstalled it and manually recreated the libraries, I didn't mind about the watch history and other stuff.
IIRC there's a post or github repo with a script to try to migrate jellyfin.

For immich you just have to copy this database files with the same command above and that's it (of course with the stack down, you don't want to copy db files while the database is running).
For the library I already had it in an external drive with a symlink, so I just had to mount it in the new machine and create a simlar symlink.

I don't run any *arr so I don't know how they'd be handled.
But I did do the migrarion of syncthing and duplicati.
For syncthing I just had to find the config path and I copied it with the same command above.
(You might need to run chown in the new machine).

For duplicati it was easier since it provides a way to export and import the configurations.

So depending on how the *arr programs handle their files it can be as easy as find their root directory and rsync it.
Maybe this could also be done for jellyfin.
Of course be sure to look for all config folders they need, some programs might split them into their working directory, into ~/.config, or ./.local, or /etc, or any other custom path.

EDIT: for jellyfin data, evaluate how hard to find is, it might be difficult, but if it's possible it doesn't require the same level of backups as your immich data, because immich normally holds data you created and can't be found anywhere else.

Most series I have them in just the main jellyfin drive.
But immich is backedup with 3-2-1, 3 copies of the data (I actually have 4), in at least 2 types of media (HDD and SSD), with 1 being offsite (rclone encrypted into e2 drive)