Yes that is absolutely something you can do. All you would need to do is set up a network share on the download box and then mount that share as a drive on the jellyfin box and then you should be able to do anything to it you would be able to do to folder sitting on its own hard drive. However, if you then use that to copy all the data to a second drive on the jellyfin system this leaves you hosting 2 copies of every file you want to see and manually renaming them over SSH or in a file explorer. But we can already access the download systems drives over the network right? So... Cant we just read all the data right off the download system over the network instead of storing two copies of it? You bet your ass we can...

What i would suggest is that you pick a single computer to be your storage system, and store ALL your data there. This will allow you to use sonarr to automatically create symlinks between the downloads and media folders, so that it can rename and organize your stuff for you without you doing anything, while also seeding the torrent in perpetuity and only using the data of a single copy of the file. For the same reason that you could use a network share to copy the data from the torrent box to the media server, you could also just use a network share to read all the data directly off a single storage box. At enterprise scale the storage is often an entirely seprate machine from the machine doing the compute, and the data all flows over network shares, allowing many compute servers to all access data from a single machine to prevent data duplication. I would strongly consider moving to a system where all your mass storage is centralized in one location, removing the need to duplicate production files across multiple machines (outside of the context of backups, which are a good thing). You can hook as many compute nodes as you want up to that single "Network attached storage" device and read the files off it to all those computers, none of which require more than a small ssd to run their os. This will give you twice as much storage on the storage node, becuase your not duplicating it on the jellyfin node.

All that being said, i am also going to assume your operating at a pretty small scale here and likely dont need very much compute, certainly not two computers worth. You should ABSOLUTELY be able to run jellyfin sonarr and qbittorrent off a single box that also acts as your NAS / central data storage location, without any performance issues, unless you have dozens of users or are transcoding like every single file your family is playing. On the jellyfin end, make sure you are set up to be direct playing the files at native resolution and not asking your server to trasncode them to another resolution, this uses lots of resources if you dont have a gpu or a intel cpu with quicksync. On the hardware side, what are you using? I run about 45 docker containers with no performance degradation and stream to up to 4 simultainous plex users WITH transcoding to lower the bitrate cuz my internet sucks and I do all that on a old 8th gen i5 i got out of the garbage. You should be able to get a used bussiness computer off ebay for 100 dollars that will run jellyfin and a sonarr stack with LOADS of overhead left, all on one device so you dont have to worry about network storage and getting bottlenecked by your networking equipment. If you tell me what it is running on i can maybe give you a better idea of if you have some kind of configuration issue, or its just antiquated hardware. But all this shit runs on a toaster. People use raspberry pis. I would imagine there is some kind of config issue leading to performance degradation.